The document describes a lesson on adding and subtracting polynomials. It includes examples of finding the degree of polynomials, adding polynomials by combining like terms, and subtracting polynomials by distributing the negative terms. The examples start simply and increase in complexity, covering the key concepts of adding and subtracting polynomials of various variables and degrees. Student practice problems are provided throughout for students to work through the skills.
The document provides notes on polynomials, including defining polynomials, describing their terms and degrees, adding and subtracting polynomials, and working through examples of finding degrees, adding, subtracting, and combining like terms of polynomials. The notes include 5 pages on adding and subtracting polynomials and working through examples step-by-step to show the process.
The document discusses several methods for adding and subtracting polynomials: using algebra tiles, the horizontal method, and the vertical method. It provides examples of adding and subtracting polynomials with one and two variables. Terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the subtracted polynomial are changed before adding.
This document discusses two methods for adding and subtracting polynomials: horizontal and vertical. For addition, terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the second polynomial are changed to addition signs before combining like terms. Both horizontal and vertical methods can be used, with horizontal grouping like terms and vertical lining them up before combining coefficients.
This document provides examples and explanations of identifying and simplifying polynomials. It begins with defining polynomials as expressions with variables that have non-negative integer exponents. Examples are provided of identifying polynomials and non-polynomials. Later pages discuss adding and subtracting polynomials by combining like terms. Worked examples are shown of simplifying polynomial expressions by removing parentheses and combining like terms. Degrees of polynomials are identified.
This document discusses adding and subtracting polynomials. It explains that to add polynomials, terms with the same variables and exponents are combined. To subtract polynomials, the opposite of the second polynomial is taken and then the polynomials are added like normal. Examples are provided of rearranging terms in decreasing exponential order, stacking like terms, and combining them to solve polynomial addition and subtraction problems.
1. The document provides examples of adding and subtracting polynomials by grouping like terms and using column form.
2. Students are asked to perform operations like adding (9y - 7x + 15a) + (-3y + 8x - 8a) and subtracting (4x^2 - 2xy + 3y^2) - (-3x^2 - xy + 2y^2).
3. The objectives are for students to learn how to add and subtract polynomials.
This document provides an overview of adding and subtracting polynomials. It defines monomials, binomials, and trinomials as polynomials with 1, 2, or 3 terms respectively. Polynomials can have any number of terms. Terms must have like variables to be combined using addition or subtraction. Examples are provided for determining the type of polynomial, simplifying polynomials by combining like terms, and adding and subtracting polynomials. Practice problems are assigned from the workbook.
The document provides notes on polynomials, including defining polynomials, describing their terms and degrees, adding and subtracting polynomials, and working through examples of finding degrees, adding, subtracting, and combining like terms of polynomials. The notes include 5 pages on adding and subtracting polynomials and working through examples step-by-step to show the process.
The document discusses several methods for adding and subtracting polynomials: using algebra tiles, the horizontal method, and the vertical method. It provides examples of adding and subtracting polynomials with one and two variables. Terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the subtracted polynomial are changed before adding.
This document discusses two methods for adding and subtracting polynomials: horizontal and vertical. For addition, terms with the same variables are combined by adding the coefficients. For subtraction, the signs of the terms in the second polynomial are changed to addition signs before combining like terms. Both horizontal and vertical methods can be used, with horizontal grouping like terms and vertical lining them up before combining coefficients.
This document provides examples and explanations of identifying and simplifying polynomials. It begins with defining polynomials as expressions with variables that have non-negative integer exponents. Examples are provided of identifying polynomials and non-polynomials. Later pages discuss adding and subtracting polynomials by combining like terms. Worked examples are shown of simplifying polynomial expressions by removing parentheses and combining like terms. Degrees of polynomials are identified.
This document discusses adding and subtracting polynomials. It explains that to add polynomials, terms with the same variables and exponents are combined. To subtract polynomials, the opposite of the second polynomial is taken and then the polynomials are added like normal. Examples are provided of rearranging terms in decreasing exponential order, stacking like terms, and combining them to solve polynomial addition and subtraction problems.
1. The document provides examples of adding and subtracting polynomials by grouping like terms and using column form.
2. Students are asked to perform operations like adding (9y - 7x + 15a) + (-3y + 8x - 8a) and subtracting (4x^2 - 2xy + 3y^2) - (-3x^2 - xy + 2y^2).
3. The objectives are for students to learn how to add and subtract polynomials.
This document provides an overview of adding and subtracting polynomials. It defines monomials, binomials, and trinomials as polynomials with 1, 2, or 3 terms respectively. Polynomials can have any number of terms. Terms must have like variables to be combined using addition or subtraction. Examples are provided for determining the type of polynomial, simplifying polynomials by combining like terms, and adding and subtracting polynomials. Practice problems are assigned from the workbook.
This document provides an overview of polynomials, including adding, subtracting, and multiplying polynomials. It begins by reviewing integer rules and exponent rules. It then covers combining like terms, adding polynomials by combining like terms and using integer addition/subtraction rules, subtracting polynomials using distribution and combining like terms, and multiplying polynomials using distribution and the box method. Examples are provided for each topic.
This document defines polynomials and describes how to perform operations on them such as addition and subtraction. It provides examples of adding and subtracting monomials and polynomials. Monomials are terms with variables and coefficients, and polynomials are the sum of monomials. Like terms refer to monomials with the same variables and exponents that can be combined. To add polynomials, like terms are lined up and their coefficients are summed. To subtract polynomials, the operation is changed to addition by using the keep-change-change method and then like terms are combined.
The document summarizes how to subtract polynomials. It explains that to subtract polynomials, one changes the sign of all terms in the subtrahend, changes the operation to addition, and then proceeds to add the polynomials by aligning like terms and combining coefficients. It provides examples of subtracting various polynomials, as well as exercises for students to practice subtracting polynomials.
Like terms are terms whose variables and exponents are the same. To add polynomials, group like terms together and add their coefficients. To multiply polynomials, use the FOIL method: First terms, Outer terms, Inner terms, Last terms.
1. The document demonstrates how to add polynomials by combining like terms.
2. Examples shown include adding polynomials with variables x, y, z, a, b, c, m, n.
3. The correct sums are found by combining coefficients of identical terms and writing the result as a single polynomial.
The document discusses multiplying polynomials, including multiplying monomials, combining like terms, and special cases such as the sum and difference of binomials, squares of binomials, and cubes of binomials. Examples are provided for multiplying polynomials with 2, 3, or 4 terms. Formulas and step-by-step workings are shown for finding products of binomial expressions.
To multiply polynomials, you can use the distributive property and properties of exponents. When multiplying monomials, group terms with the same bases and add their exponents. When multiplying binomials, use FOIL or distribute one binomial over the other. For polynomials with more than two terms, you can distribute or use a rectangle model to systematically multiply each term.
The document discusses three methods for multiplying polynomials: the distributive property, FOIL (First, Outer, Inner, Last), and the box method. It provides examples of multiplying polynomials using each method. The key steps of FOIL are to multiply the first, outer, inner, and last terms of each binomial being multiplied. The box method involves drawing a box and writing one polynomial above and beside the box before multiplying the terms. The document emphasizes that all three methods will provide the same answer when multiplying polynomials.
The student will learn to:
1. Add and subtract polynomials by grouping like terms.
2. Use column form to add and subtract polynomials with three terms.
3. Simplify polynomial expressions involving addition and subtraction.
This document discusses two methods for adding and subtracting polynomials:
1) The horizontal method involves grouping like terms together and keeping the signs with each term.
2) The vertical method lines up like terms and keeps the signs with each term.
To subtract polynomials, change all the signs in the second set and then add the polynomials as if it were an addition problem. Either the horizontal or vertical method can be used, depending on how the problem is laid out.
The document describes three methods for multiplying polynomials:
1) The distributive property, which involves multiplying each term of one polynomial with each term of the other.
2) FOIL (First, Outer, Inner, Last), which is a mnemonic for multiplying binomials by multiplying corresponding terms.
3) The box method, which involves drawing a box and writing one polynomial above and beside the box, then multiplying corresponding terms. Examples are provided to demonstrate each method.
Adding and subtracting polynomials involves combining like terms. Like terms are terms that have the same variables and the same exponents. To add polynomials, terms are grouped by their like terms and the coefficients are combined by adding them. To subtract polynomials, the process is the same as adding the opposite of the second polynomial. The opposite of each term is found by changing its sign. Then the like terms are combined in the same way as when adding.
The document provides examples and explanations of operations involving polynomials, including:
1) Adding polynomials by combining like terms such as 5x + 3x and finding the sum of polynomials using tiles.
2) Performing addition, subtraction, and multiplication of polynomials with various terms.
3) Dividing polynomials using tiles to represent the division operation and finding quotients and remainders.
To add and subtract polynomials:
1) Add like terms by combining coefficients of identical terms
2) Sort polynomials in descending order by exponent before adding or subtracting
3) To subtract polynomials, change the sign of the second polynomial and add it to the first
The student will be able to:
1. Add and subtract polynomials.
The document provides examples of adding and subtracting polynomials by combining like terms. It includes adding and subtracting polynomials using column form. There are also practice problems for students to find the sum or difference of polynomials.
Addition and subtraction of polynomialsjesus abalos
The document provides information about adding and subtracting algebraic expressions:
- Like terms are algebraic expressions with the same variables and exponents.
- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
To subtract polynomials, you keep the sign of the first term, change subtraction to addition, and flip the sign of the second term. You then apply this process to every term in the polynomials. The document provides an example rule, two practice problems to try, and the answers to check your work.
This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
1. The document provides instructions and examples for a math lesson on adding and subtracting polynomials.
2. Students are assigned problems from their textbook and a checkpoint due the following Tuesday.
3. The lesson examples show how to add and subtract polynomials by combining like terms in either a horizontal or vertical format and expressing the solution in standard form.
This document provides an overview of polynomials, including adding, subtracting, and multiplying polynomials. It begins by reviewing integer rules and exponent rules. It then covers combining like terms, adding polynomials by combining like terms and using integer addition/subtraction rules, subtracting polynomials using distribution and combining like terms, and multiplying polynomials using distribution and the box method. Examples are provided for each topic.
This document defines polynomials and describes how to perform operations on them such as addition and subtraction. It provides examples of adding and subtracting monomials and polynomials. Monomials are terms with variables and coefficients, and polynomials are the sum of monomials. Like terms refer to monomials with the same variables and exponents that can be combined. To add polynomials, like terms are lined up and their coefficients are summed. To subtract polynomials, the operation is changed to addition by using the keep-change-change method and then like terms are combined.
The document summarizes how to subtract polynomials. It explains that to subtract polynomials, one changes the sign of all terms in the subtrahend, changes the operation to addition, and then proceeds to add the polynomials by aligning like terms and combining coefficients. It provides examples of subtracting various polynomials, as well as exercises for students to practice subtracting polynomials.
Like terms are terms whose variables and exponents are the same. To add polynomials, group like terms together and add their coefficients. To multiply polynomials, use the FOIL method: First terms, Outer terms, Inner terms, Last terms.
1. The document demonstrates how to add polynomials by combining like terms.
2. Examples shown include adding polynomials with variables x, y, z, a, b, c, m, n.
3. The correct sums are found by combining coefficients of identical terms and writing the result as a single polynomial.
The document discusses multiplying polynomials, including multiplying monomials, combining like terms, and special cases such as the sum and difference of binomials, squares of binomials, and cubes of binomials. Examples are provided for multiplying polynomials with 2, 3, or 4 terms. Formulas and step-by-step workings are shown for finding products of binomial expressions.
To multiply polynomials, you can use the distributive property and properties of exponents. When multiplying monomials, group terms with the same bases and add their exponents. When multiplying binomials, use FOIL or distribute one binomial over the other. For polynomials with more than two terms, you can distribute or use a rectangle model to systematically multiply each term.
The document discusses three methods for multiplying polynomials: the distributive property, FOIL (First, Outer, Inner, Last), and the box method. It provides examples of multiplying polynomials using each method. The key steps of FOIL are to multiply the first, outer, inner, and last terms of each binomial being multiplied. The box method involves drawing a box and writing one polynomial above and beside the box before multiplying the terms. The document emphasizes that all three methods will provide the same answer when multiplying polynomials.
The student will learn to:
1. Add and subtract polynomials by grouping like terms.
2. Use column form to add and subtract polynomials with three terms.
3. Simplify polynomial expressions involving addition and subtraction.
This document discusses two methods for adding and subtracting polynomials:
1) The horizontal method involves grouping like terms together and keeping the signs with each term.
2) The vertical method lines up like terms and keeps the signs with each term.
To subtract polynomials, change all the signs in the second set and then add the polynomials as if it were an addition problem. Either the horizontal or vertical method can be used, depending on how the problem is laid out.
The document describes three methods for multiplying polynomials:
1) The distributive property, which involves multiplying each term of one polynomial with each term of the other.
2) FOIL (First, Outer, Inner, Last), which is a mnemonic for multiplying binomials by multiplying corresponding terms.
3) The box method, which involves drawing a box and writing one polynomial above and beside the box, then multiplying corresponding terms. Examples are provided to demonstrate each method.
Adding and subtracting polynomials involves combining like terms. Like terms are terms that have the same variables and the same exponents. To add polynomials, terms are grouped by their like terms and the coefficients are combined by adding them. To subtract polynomials, the process is the same as adding the opposite of the second polynomial. The opposite of each term is found by changing its sign. Then the like terms are combined in the same way as when adding.
The document provides examples and explanations of operations involving polynomials, including:
1) Adding polynomials by combining like terms such as 5x + 3x and finding the sum of polynomials using tiles.
2) Performing addition, subtraction, and multiplication of polynomials with various terms.
3) Dividing polynomials using tiles to represent the division operation and finding quotients and remainders.
To add and subtract polynomials:
1) Add like terms by combining coefficients of identical terms
2) Sort polynomials in descending order by exponent before adding or subtracting
3) To subtract polynomials, change the sign of the second polynomial and add it to the first
The student will be able to:
1. Add and subtract polynomials.
The document provides examples of adding and subtracting polynomials by combining like terms. It includes adding and subtracting polynomials using column form. There are also practice problems for students to find the sum or difference of polynomials.
Addition and subtraction of polynomialsjesus abalos
The document provides information about adding and subtracting algebraic expressions:
- Like terms are algebraic expressions with the same variables and exponents.
- Unlike terms cannot be combined.
- To add algebraic expressions, combine like terms by adding the coefficients.
- To subtract expressions, change the sign of the second expression and then add as if adding.
To subtract polynomials, you keep the sign of the first term, change subtraction to addition, and flip the sign of the second term. You then apply this process to every term in the polynomials. The document provides an example rule, two practice problems to try, and the answers to check your work.
This document contains examples of multiplying, expanding, and simplifying rational expressions. Some examples involve breaking rational expressions into sums and differences of fractions. Other examples use long division to write rational expressions in the form of a quotient plus a remainder over the divisor. The rational expressions involve variables and operations.
This document discusses multiplying polynomials. It begins with examples of multiplying monomials by using the properties of exponents. It then covers multiplying a polynomial by a monomial using the distributive property. Examples are provided for multiplying binomials by binomials using both the distributive property and FOIL method. The document concludes by explaining methods for multiplying polynomials with more than two terms, such as using the distributive property multiple times, a rectangle model, or a vertical method similar to multiplying whole numbers.
1. The document provides instructions and examples for a math lesson on adding and subtracting polynomials.
2. Students are assigned problems from their textbook and a checkpoint due the following Tuesday.
3. The lesson examples show how to add and subtract polynomials by combining like terms in either a horizontal or vertical format and expressing the solution in standard form.
This musical composition is made up of repeating phrases of notes and chords played over four pages. It utilizes the notes A, B, C, and Q throughout, along with symbols like question marks, numbers, and atypical musical notation. The piece cycles through blending and varying these elements in a non-traditional structure across its full text.
This document provides examples of calculating slope from points on a line. It defines slope as the rise over the run and gives the formula m = (y2 - y1) / (x2 - x1). Several examples show applying this formula to find the slope of a line given two points it passes through. One example calculates the average rate of change in temperature over time. Another finds the x-value that gives a line between two points a specific slope. The last example calculates the average yearly increase in radio stations from 1980 to 1994.
This document provides instructions for using the Acunetix Web Vulnerability Scanner to scan websites for security vulnerabilities. It outlines the 5 step scan wizard process, including selecting targets, confirming technologies detected, specifying crawler options, choosing a scanning profile and mode, and configuring logins for password protected areas. The steps guide the user through launching a scan, with notes on authorization requirements and optimizing the scan based on detected website technologies.
Retail store connect technical referenceAhmed Farag
- Retail Store Connect is a service that replicates data between head office and store databases to keep them synchronized.
- It can be configured in different ways, such as having a "forwarder" instance that passes packages to other Retail Store Connect instances for load balancing.
- The Retail Store Connect Settings tool is used to configure instances of the Retail Store Connect service, specifying properties like the server name and database connections.
How has COMNet involved Pashtoon Women in Quetta for PEIgedrosia
1) Women in Pashtoonabad gather at bazaars, health centers, girls' schools, vocational centers, and madrassas. Influential women who could encourage polio vaccination include mothers, female teachers, caregivers, and religious elders.
2) To involve women in polio eradication, social mobilizers conducted women-to-women awareness sessions by engaging influential females. They also held polio awareness sessions at educational institutions, skill development centers, and religious schools.
3) Key vaccination messages were disseminated at marketplaces through loudspeakers, flyers, and awareness sessions at health centers where women sought medical care for themselves and their children. These efforts led
Charlie's uncle emerged from the shadows and stood in front of him. Initially, escaping from a dead end was just a means of avoiding the situation. That day was the beginning of Charlie's troubles. Charlie managed to dodge his uncle's attacks as his uncle's movements were clumsy due to alcohol.
The document discusses different business opportunities through a mobile phone franchise called Vmobile. It outlines how individuals can start as retailers or dealers to earn commissions from selling prepaid phone and cable packages. As retailers, people can sign up other retailers and earn overrides on their sales. As dealers, individuals can endorse other dealers and retailers and earn bonuses on all of their downstream sales networks over 6 months. Even just endorsing a few others can generate over ₱76,000 in income, more than the initial ₱4,000 capital required.
Livehelp server user guide for wordpressActiveHelper
The document outlines the 4-step process for configuring a LiveHelp server, including 1) configuring domains, 2) adding and managing agents, 3) configuring tracking widgets, and 4) agents signing in using the Support Panel desktop or mobile app to go online. Key steps include adding domains and agents, generating widgets or scripts, and agents signing in through the Support Panel to change their status to online and begin receiving chats.
1) If line 1 has a slope of 3, the slope of a perpendicular line 2 needs to be -1/3.
2) If a line has the equation x + 4y = 7, a parallel line will have a slope of -1/4.
3) If a line has the equation x + 4y = 7, a perpendicular line will have a slope of 4.
The document discusses order of operations and provides examples of solving expressions and equations using the proper order of operations. It also includes two word problems, one about predicting population decrease over time using an exponential function, and another about money added to a bank account each week.
The document describes safety cabinets from the CLASSIC line for storing hazardous substances. The CLASSIC XL model has shelves or shelves and a center partition wall. It is 1195 mm wide and tested according to European safety standards. The document provides details on the construction, fittings, colors, and accessories available for the safety cabinets.
The document discusses multiplying and dividing variable expressions. It provides examples of simplifying expressions using the distributive property and the property of the opposite of a sum. It also demonstrates dividing variable expressions by writing the division as a fraction and simplifying. Key steps include distributing terms, dividing each term in the numerator by the denominator, and evaluating expressions for given variable values.
PMR Form 3 Mathematics Algebraic FractionsSook Yen Wong
The document provides instructions for expanding and factorizing algebraic expressions involving single and double brackets. It explains how to expand brackets by distributing terms inside brackets to each term outside. For factorizing, it describes finding common factors and grouping terms. It also covers techniques for factorizing quadratic expressions, difference of squares, and grouping. Further sections cover simplifying algebraic fractions through factorizing numerators and denominators and combining like terms.
This document provides examples and instructions for subtracting polynomials. It begins with examples of subtracting terms like x+7 and 3x+9. It then shows how to subtract polynomials by removing common terms and combining like terms, such as (3x^2 + x + 2) - (2x^2 - x + 3). The document ends with homework assignments involving subtracting various polynomials.
This document provides examples and explanations for multiplying polynomials by monomials. It begins with examples of writing the name and degree of polynomials. It then provides examples of multiplying polynomials together through combining like terms. The document reinforces the concept with additional examples and problems for students to work through. It concludes by asking a question about how the laws of exponents apply to this topic.
Rational expressions are fractions where the numerator and denominator are polynomials. To simplify rational expressions, we first factor the polynomials and then cancel any common factors. Adding and subtracting rational expressions follows the same process as fractions - find the least common denominator, multiply the numerators and denominators to get the same denominator, then add or subtract the numerators. Multiplying rational expressions involves factoring and cancelling common factors between the numerator and denominator. To divide rational expressions, we multiply the first expression by the reciprocal of the second expression and then factor and cancel. Word problems involving rational expressions can be solved by identifying what is known and unknown, setting up equations relating the known and unknown values, and then solving the equations.
Here are the steps to solve problem #1 on page 74:
1) Simplify the expression: -3(x - 5)
2) Use the property that anything inside the parentheses will be opposite if there is a negative sign outside: -3(x - 5) = -3x + 15
3) Simplify: -3x + 15
The simplified expression is: -3x + 15
Module 9 Topic 2 multiplying polynomials - part 1Lori Rapp
The document discusses multiplying polynomials by monomials. It explains that to multiply a polynomial by a monomial, each term inside the polynomial parentheses should be multiplied by the monomial. This will preserve the number of terms. Examples are provided to demonstrate distributing the monomial to each term and then multiplying the terms. Readers are instructed to work through practice problems in their notebook and check their work.
This document provides information and examples on multiplying polynomials, including:
1) Multiplying a monomial and polynomial using the distributive property.
2) Multiplying two polynomials using both the horizontal and vertical methods.
3) Factoring trinomials and identifying similar and conjugate binomials. Methods like FOIL and grouping are discussed.
The document discusses multiplying polynomials by monomials. It explains that to multiply a polynomial by a monomial, you distribute the monomial to each term inside the parentheses. This is done by multiplying each term by the monomial. The number of terms after multiplying will be the same as the number of terms inside the original parentheses. It provides examples of multiplying different polynomials by monomials. It has students work through examples on their own and check their work.
The document shows examples of advanced factoring techniques including factoring by grouping, factoring quadratics, and factoring a polynomial into linear factors in order to graph the solution set. It provides step-by-step workings to factor expressions into their simplest linear factors in order to determine possible solutions for equations.
The document contains a table matching pairs of quadratic equations that have been factorized. There are 16 pairs of equations in total. The factorizations include expressions of the form (ax + b)(cx + d) where a, b, c, and d are coefficients.
The key words provided summarize the main concepts needed to factorize quadratic equations, including the terms quadratic, solve, coefficient, factorize, hence, factors, and brackets.
8-5 Adding and Subtracting Rational Expressionsrfrettig
1) Rational functions can be added or subtracted if they have a common denominator. To find the least common denominator (LCD), multiply the individual denominators together and divide by their greatest common factor (GCF).
2) Examples are provided of finding the LCD of rational expressions and adding or subtracting rational expressions after finding the LCD. Factoring is used to find the LCD.
3) The document provides examples of adding, subtracting, and simplifying rational expressions by finding the LCD and distributing terms in the numerators. Practice problems are assigned from the textbook.
1. The document contains 10 math word problems with solutions.
2. The problems involve solving linear equations for variables like x and a.
3. Steps shown include isolating the variable, combining like terms, and evaluating.
This document contains the marking scheme for the Additional Mathematics trial SPM 2009 paper 1. It provides the full workings and marks for each question. The key points assessed include algebraic manipulation, logarithmic and trigonometric functions, vectors, and statistics such as variance. In total there are 22 questions on topics commonly found in Additional Mathematics exams.
This document provides examples of multiplying monomials and polynomials. It begins with an anticipatory set asking students to find the area of x3(x-4). It then defines multiplying polynomials by monomials. The document provides 6 examples of multiplying monomials by polynomials. It revisits the anticipatory set and provides 2 more examples. It ends by asking how the laws of exponents apply to this topic.
This document discusses integration by substitution. It explains that instead of calling the function of x "STUFF", it will be called u, and its derivative will be du. The reader is instructed to identify part of the integrand as u and part as du. Examples are provided of using this technique to integrate functions of the form u^n, including (x^2 + 1)^3 as an example. The reader is prompted to try integrating (3x - 9)/(9x^2) and other functions using this substitution method.
The document discusses synthetic division and factoring polynomials. It provides an example of using synthetic division to divide one polynomial by another. Key points covered include: the remainder theorem, which states that the remainder of dividing a polynomial P(x) by (x-c) is equal to the value of P(c); and the factor theorem, which states that c is a zero of P(x) if and only if (x-c) is a factor of P(x). The document then provides an example of using the fact that P(1)=0 to factor a polynomial completely.
Module 10 Topic 4 solving quadratic equations part 1Lori Rapp
This document provides instruction on solving quadratic equations. It begins with an introduction explaining why quadratic equations are useful and includes a video example. It then defines quadratic equations and provides examples of solving quadratic equations by factoring and using the quadratic formula. The document uses examples and "Try It" problems for students to practice each step in solving quadratic equations.
This document provides examples of subtracting polynomials. It begins with examples of subtracting terms like x+7 and 3x+9. It then shows how to subtract polynomials by removing corresponding terms and using the opposite sign. For example, (3x^2 + x + 2) - (2x^2 - x + 3) is solved by removing the 2x^2 terms and changing the sign of the remaining terms to get x^2 + x - 1. It provides several more examples and problems for students to solve, including finding the area of a shaded region between a rectangle and square using polynomial subtraction.
1) Simultaneous equations involve two variables in two equations that are solved simultaneously to find the values of the variables.
2) To solve simultaneous equations, one first expresses one variable in terms of the other by changing the subject of one linear equation, then substitutes this into the other equation to obtain a quadratic equation.
3) This quadratic equation is then solved using factorisation or the quadratic formula to find the values of the variables that satisfy both original equations.
Similar to Topic 1 adding & subtracting polynomials (20)
1. There are three main methods of heat transfer: conduction, convection, and radiation.
2. The Earth is heated through radiation from the sun, with some energy being absorbed, some reflected, and some transmitted through substances like water and air.
3. Factors like land/water distribution, altitude, latitude, and cloud cover contribute to geographic and climatic differences in surface temperatures around the world.
Air pressure is exerted equally in all directions by the air around us. It can be measured using a barometer, with units of mmHg or atm. Rapid changes in air pressure often indicate approaching weather systems - falling pressure means storms while rising pressure means clearing weather. Winds are created by differences in air pressure between areas. Global wind patterns are caused by uneven heating of the Earth, with the hottest air at the equator rising and causing circulation of air masses. High and low pressure systems form based on temperature and moisture levels, influencing wind direction.
1. There are three main methods of heat transfer: conduction, convection, and radiation.
2. The Earth is heated through radiation from the sun, with some energy being absorbed, some reflected, and some transmitted through substances like water and air.
3. Temperatures vary based on factors like proximity to water vs. land, altitude, latitude, and cloud cover, with land experiencing greater temperature ranges than water and higher latitudes being cooler on average.
Air pressure is exerted equally in all directions by the air around us. It can be measured using a barometer, with units of mmHg or atm. Rapid changes in air pressure often indicate approaching weather systems - falling pressure means storms while rising pressure means clearing weather. Winds are created by differences in air pressure between areas. Global wind patterns are caused by uneven heating of the Earth, with the hottest air at the equator rising and causing circulation of air masses. High and low pressure systems form based on temperature and moisture levels, influencing wind direction.
The document provides 20 math word problems that require solving equations for a variable. It works through each problem step-by-step showing the work and reasoning to isolate the variable. Common steps include distributing terms, combining like terms, adding/subtracting the same quantity to both sides of the equation, and dividing/multiplying both sides by the same non-zero quantity. The final problems involve multiple steps to solve more complex equations.
The atmosphere is divided into four main layers - the troposphere, stratosphere, mesosphere, and thermosphere. Each layer is defined by changes in temperature. The troposphere extends from Earth's surface to around 12 km and contains around 90% of the atmosphere. The stratosphere extends from 12-50 km and contains the ozone layer which absorbs harmful UV radiation. Air is composed primarily of nitrogen (78%) and oxygen (21%).
Maps have evolved over time to become more accurate representations of the world. Early maps from 4000 BC showed where people lived and tax boundaries, while Ptolemy created the first world map in 150 AD using Greek and Roman cartography. In the Middle Ages, maps reflected cultural beliefs and guided pilgrims. Modern technology like GPS and GIS has led to highly accurate digital maps through satellite imagery and data analysis.
This document provides an overview of current environmental issues facing the Earth. It discusses how human population growth and industrialization have negatively impacted the environment through increased consumption of non-renewable resources, pollution, and loss of biodiversity. As human societies developed from hunter-gatherers to agricultural to industrial, their demands on the environment expanded and dependence on fossil fuels rose, resulting in problems like pollution and depletion of resources. The core issues discussed are overconsumption, pollution, and biodiversity loss due to rising human populations. The document also introduces the concept of sustainability and the goal of creating a sustainable society that does not exceed the Earth's limits.
The document discusses water resources and water treatment. It explains that most of Earth's water is ocean water, while freshwater comes from surface water, groundwater, and the water cycle. It then describes how surface water forms rivers and lakes, groundwater forms aquifers, and how water is used residentially, industrially, and agriculturally. Issues like overdraft and habitat degradation are discussed, along with water treatment processes like desalination, purification through screening, sedimentation, filtration, aeration, and sterilization.
The document summarizes different types of telescopes and space probes used to explore the universe, and some of their key contributions. It discusses radio, infrared, visible light, ultraviolet, X-ray, and gamma ray telescopes, as well as notable space telescopes like Hubble and the upcoming James Webb Space Telescope. It also outlines various space probes sent to study planets, moons, comets and more, providing images to advance scientists' understanding of the solar system. Finally, it lists some technologies that were developed for space exploration but later found applications on Earth.
1. Stars form from dense clouds of gas and dust called nebulae when gravity overcomes pressure, causing the cloud to collapse and heat up. Nuclear fusion then begins, forming a star.
2. Stars exist in different stages depending on their mass. Medium stars become red giants then white dwarfs. Massive stars explode as supernovae, leaving neutron stars or black holes.
3. Astronomers use tools like spectroscopy, star charts, and parallax to determine properties of stars like composition, temperature, brightness, and distance. This helps classify stars and map their life cycles.
The document discusses planetary motion and the movement of celestial bodies. It describes how early astronomers like Ptolemy, Copernicus, and Kepler developed models of the solar system. Johannes Kepler developed his three laws of planetary motion, establishing that planets orbit the sun in ellipses. Isaac Newton later explained that gravity is the force that causes planets to orbit the sun. The document also discusses the Earth's rotation, revolution, and axial tilt, which cause seasons. It describes the moon's orbit and how its phases, tides, and eclipses occur due to the positions of the Earth, moon, and sun.
The document summarizes theories about the formation and evolution of the universe:
1) The Big Bang theory proposes that approximately 13.8 billion years ago, all matter in the observable universe was condensed into an incredibly dense singularity that suddenly exploded, causing the universe to expand rapidly and form light elements.
2) As the universe expanded, gravity caused matter to condense into the first stars and galaxies. Observations of redshifts in light from distant galaxies provide evidence the universe continues expanding today.
3) Current theories suggest three possible ultimate fates for the universe depending on its overall density - continued expansion, eventual contraction, or a stable flat state.
Chapter 18 – air pollution and global changesAnnie cox
This document discusses various types of air pollution and their local and global effects. It describes how local pollution from activities like burning fossil fuels can cause issues worldwide through mechanisms like acid rain, ozone depletion, and global warming. Five major types of outdoor air pollutants are discussed - particulates, oxides, photochemical smog, hydrocarbons, and CFCs. The document also notes how air pollution affects human health, other organisms, and causes global problems through acid deposition, ozone layer thinning, and climate change due to increased greenhouse gases.
Meteorologists use various data sources to predict the weather, but it is difficult to always be correct due to rapidly changing atmospheric conditions. Data comes from surface weather stations, weather balloons, satellites, radars, and computer models. Air masses and fronts influence weather, and storms form when different air masses meet. Hurricanes require specific conditions like warm ocean water. While meteorology studies the atmosphere, tiny differences in data can lead to different model results, so multiple models are used to predict daily weather.
1. The document discusses the four layers of Earth's atmosphere and how factors like temperature, pressure, wind, and moisture interact to cause weather.
2. It explains that the lowest layer, the troposphere, contains almost all of the atmosphere's mass and water and is where weather occurs.
3. The stratosphere contains the protective ozone layer. The thermosphere is where satellites orbit and temperatures can reach 2000 degrees Celsius.
4. Weather results from interactions of heat, air pressure, wind, and moisture. Unequal heating causes pressure differences that create wind, and wind circulations around the globe.
The document discusses various types of water pollution including chemical, thermal, radioactive, sediments, and sewage. The largest sources of water pollution are agricultural runoff and sewage. Developing countries often lack proper sewage treatment facilities. Water pollution has been an issue for centuries but legislation and enforcement have been weak, allowing the problem to continue. The top source of water pollution in North Carolina is stormwater runoff carrying various pollutants. Solutions include limiting development near waterways and preserving wetlands to naturally filter water.
The document discusses water resources and water treatment. It explains that most of Earth's water is ocean water, while freshwater comes from surface water, groundwater, and the water cycle. It then describes how surface water flows in rivers and collects in lakes and reservoirs, and how groundwater flows through permeable rock layers and can be accessed through wells. The document also outlines water use for residential, industrial, and agricultural purposes, as well as problems like overdraft and habitat degradation. It concludes by explaining processes for desalination and water purification to remove salts and contaminants, making water safe for drinking.
The document summarizes how oceans shape our lives by discussing the distribution of water on Earth, properties of ocean water, ocean motions, shoreline formations, and the barrier islands of North Carolina's Outer Banks. It explains that barrier islands formed due to rising sea levels, a large sand supply, a gently sloping coast, and wave energy. Humans are increasing the rate of barrier island migration through armoring and development, which reduces available sand and increases erosion risks. Ultimately, one cannot have buildings and beaches in the long run due to the dynamic coastal environment.
Double-sided equations can be solved by doing the same operation to both sides of the equals sign. This keeps the equation balanced. Terms should be combined so that variables are grouped together on one side and constants on the other. When a term crosses the equals sign to the other side, its sign changes. By following these steps, variables can be isolated so the equation can be solved.
2. What is a Polynomial?
View this Cool Math lesson introducing
Polynomials.
There is only the 1 page of notes.
3. Describe Polynomials
View this Cool Math lesson introducing
terms describing Polynomials.
There are 3 pages of notes.
Be sure to complete the Your Turn problems
in your notebook. Check your answers on
the next slides.
5. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
6. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
Find the degree of a 4b - 7a 2b2 .
7. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
Find the degree of a 4b - 7a 2b2 .
The degree is 5.
8. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
Find the degree of a 4b - 7a 2b2 .
The degree is 5.
Find the degree of 7x - 1.
9. Your Turn on Page 3
Find the degree of 3x - x2 .
The degree is 2.
Find the degree of a 4b - 7a 2b2 .
The degree is 5.
Find the degree of 7x - 1.
The degree is 1.
10. Adding & Subtracting
Polynomials
View this Cool Math lesson to learn how to
add and subtract polynomials.
There are 5 pages of notes.
Be sure to complete the Try It and Your
Turn problems in your notebook. Check
your answers on the next slides.
11. Try It on Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
12. Try It on Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
13. Try It on Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
14. Try It on Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
Rearrange so
like terms are
together.
15. Try It on Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
Rearrange so
like terms are
together. ( 3x + 6x ) + ( −4w + 2w ) + (1z + 5z )
16. Try It on Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
Rearrange so
like terms are
together. ( 3x + 6x ) + ( −4w + 2w ) + (1z + 5z )
Combine like terms.
17. Try It on Page 2
Simplify: ( 3x − 4w + z ) + ( 6x + 2w + 5z )
Remove parenthesis.
3x − 4w + z + 6x + 2w + 5z
Rearrange so
like terms are
together. ( 3x + 6x ) + ( −4w + 2w ) + (1z + 5z )
Combine like terms.
9x − 2w + 6z
18. Your Turn on Page 3
Simplify. ( 8x 2
) ( )
− x + 3 + 2x + 6x + 1
2
19. Your Turn on Page 3
Simplify. ( 8x 2
) ( )
− x + 3 + 2x + 6x + 1
2
Remove parenthesis.
20. Your Turn on Page 3
Simplify. ( 8x 2
) ( )
− x + 3 + 2x + 6x + 1
2
Remove parenthesis.
8x − x + 3 + 2x + 6x + 1
2 2
21. Your Turn on Page 3
Simplify. ( 8x 2
) ( )
− x + 3 + 2x + 6x + 1
2
Remove parenthesis.
8x − x + 3 + 2x + 6x + 1
2 2
Rearrange so like
terms are together.
22. Your Turn on Page 3
Simplify. ( 8x 2
) (
− x + 3 + 2x + 6x + 1
2
)
Remove parenthesis.
8x − x + 3 + 2x + 6x + 1
2 2
Rearrange so like
terms are together.
( 8x 2
)
+ 2x + ( −x + 6x ) + ( 3 + 1)
2
23. Your Turn on Page 3
Simplify. ( 8x 2
) (
− x + 3 + 2x + 6x + 1
2
)
Remove parenthesis.
8x − x + 3 + 2x + 6x + 1
2 2
Rearrange so like
terms are together.
( 8x 2
)
+ 2x + ( −x + 6x ) + ( 3 + 1)
2
Combine like terms.
24. Your Turn on Page 3
Simplify. ( 8x 2
) (
− x + 3 + 2x + 6x + 1
2
)
Remove parenthesis.
8x − x + 3 + 2x + 6x + 1
2 2
Rearrange so like
terms are together.
( 8x 2
)
+ 2x + ( −x + 6x ) + ( 3 + 1)
2
Combine like terms.
10x + 5x + 4
2
25. Try It on Page 3
Simplify. ( 4a b − ab + 3b
2 2
) (
− 1 + a b + 8ab − 6a + 5
2 2
)
26. Try It on Page 3
Simplify. ( 4a b − ab + 3b
2 2
) (
− 1 + a b + 8ab − 6a + 5
2 2
)
Remove parenthesis.
27. Try It on Page 3
Simplify. ( 4a b − ab + 3b
2 2
) (
− 1 + a b + 8ab − 6a + 5
2 2
)
Remove parenthesis.
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
2 2 2 2
28. Try It on Page 3
Simplify. ( 4a b − ab + 3b
2 2
) (
− 1 + a b + 8ab − 6a + 5
2 2
)
Remove parenthesis.
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
2 2 2 2
Rearrange so like terms are together.
29. Try It on Page 3
Simplify. ( 4a b − ab + 3b
2 2
) (
− 1 + a b + 8ab − 6a + 5
2 2
)
Remove parenthesis.
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
2 2 2 2
Rearrange so like terms are together.
( 4a b + a b ) + ( −ab + 8ab ) + 3b
2 2 2
− 6a + ( −1 + 5 )
2
30. Try It on Page 3
Simplify. ( 4a b − ab + 3b
2 2
) (
− 1 + a b + 8ab − 6a + 5
2 2
)
Remove parenthesis.
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
2 2 2 2
Rearrange so like terms are together.
( 4a b + a b ) + ( −ab + 8ab ) + 3b
2 2 2
− 6a + ( −1 + 5 )
2
Combine like terms.
31. Try It on Page 3
Simplify. ( 4a b − ab + 3b
2 2
) (
− 1 + a b + 8ab − 6a + 5
2 2
)
Remove parenthesis.
4a b − ab + 3b − 1 + a b + 8ab − 6a + 5
2 2 2 2
Rearrange so like terms are together.
( 4a b + a b ) + ( −ab + 8ab ) + 3b
2 2 2
− 6a + ( −1 + 5 )
2
Combine like terms.
5a b + 7ab + 3b − 6a + 4
2 2 2
32. Your Turn on Page 4
Simplify. ( 3x 2
) (
− 4x + 8 − x + 7x − 5
2
)
33. Your Turn on Page 4
Simplify. ( 3x 2
) (
− 4x + 8 − x + 7x − 5
2
)
Remove parenthesis but
distribute the negative.
34. Your Turn on Page 4
Simplify. ( 3x 2
) (
− 4x + 8 − x + 7x − 5
2
)
Remove parenthesis but
distribute the negative.
3x − 4x + 8 − x − 7x + 5
2 2
35. Your Turn on Page 4
Simplify. ( 3x 2
) (
− 4x + 8 − x + 7x − 5
2
)
Remove parenthesis but
distribute the negative.
3x − 4x + 8 − x − 7x + 5
2 2
Rearrange so like
terms are together.
36. Your Turn on Page 4
Simplify. ( 3x 2
) (
− 4x + 8 − x + 7x − 5 2
)
Remove parenthesis but
distribute the negative.
3x − 4x + 8 − x − 7x + 5
2 2
Rearrange so like
terms are together.
( 3x 2
−x 2
) + ( −4x − 7x ) + ( 8 + 5 )
37. Your Turn on Page 4
Simplify. ( 3x 2
) (
− 4x + 8 − x + 7x − 5 2
)
Remove parenthesis but
distribute the negative.
3x − 4x + 8 − x − 7x + 5
2 2
Rearrange so like
terms are together.
( 3x 2
−x 2
) + ( −4x − 7x ) + ( 8 + 5 )
Combine like terms.
38. Your Turn on Page 4
Simplify. ( 3x 2
) (
− 4x + 8 − x + 7x − 5 2
)
Remove parenthesis but
distribute the negative.
3x − 4x + 8 − x − 7x + 5
2 2
Rearrange so like
terms are together.
( 3x 2
−x 2
) + ( −4x − 7x ) + ( 8 + 5 )
Combine like terms.
2x − 11x + 13
2
39. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
40. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
Write as a subtraction.
41. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
Write as a subtraction.
(x 2 2
) (
− 4xy + 5y − 3x − 6xy + y
2 2
)
42. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
Write as a subtraction.
(x 2 2
) (
− 4xy + 5y − 3x − 6xy + y
2 2
)
Remove parenthesis but
distribute the negative.
43. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
Write as a subtraction.
(x 2 2
) (
− 4xy + 5y − 3x − 6xy + y
2 2
)
Remove parenthesis but
distribute the negative.
x − 4xy + 5y − 3x + 6xy − y
2 2 2 2
44. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
Write as a subtraction.
(x 2 2
) (
− 4xy + 5y − 3x − 6xy + y
2 2
)
Remove parenthesis but
distribute the negative.
x − 4xy + 5y − 3x + 6xy − y
2 2 2 2
Rearrange so like
terms are together.
45. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
Write as a subtraction.
(x 2 2
) (
− 4xy + 5y − 3x − 6xy + y
2 2
)
Remove parenthesis but
distribute the negative.
x − 4xy + 5y − 3x + 6xy − y
2 2 2 2
Rearrange so like
terms are together.
(x 2 2
)
− 3x + ( −4xy + 6xy ) + 5y − y( 2 2
)
46. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
Write as a subtraction.
(x 2 2
) (
− 4xy + 5y − 3x − 6xy + y
2 2
)
Remove parenthesis but
distribute the negative.
x − 4xy + 5y − 3x + 6xy − y
2 2 2 2
Rearrange so like
terms are together.
(x 2 2
)
− 3x + ( −4xy + 6xy ) + 5y − y( 2 2
)
Combine like terms.
47. Your Turn on Page 5
Subtract 3x − 6xy + y from x − 4xy + 5y .
2 2 2 2
Write as a subtraction.
(x 2 2
) (
− 4xy + 5y − 3x − 6xy + y 2 2
)
Remove parenthesis but
distribute the negative.
x − 4xy + 5y − 3x + 6xy − y
2 2 2 2
Rearrange so like
terms are together.
(x 2 2
)
− 3x + ( −4xy + 6xy ) + 5y − y ( 2 2
)
Combine like terms.
−2x + 2xy + 4y
2 2
48. Gizmo Practice Activity
Let's practice some more at the Explore Learning
website using the username and password given to
you AND found in the support documents.
Search for the "Addition of Polynomials - Activity
A" Gizmo, launch the Gizmo and try at least 5
problems.
After you have completed at least 5 Gizmo problems,
scroll down to the Assessment Questions and
complete the 4 multiple choice problems.
Pronto or Message me if you have questions.
49. Adding & Subtracting
Polynomials Practice
Practice Adding & Subtracting Polynomials at this Cool
Math website.
Select the “Give me a Problem” button to keep trying
problems.
Do your work in a notebook.
When you select “What’s the Answer?” your answer is
erased and correct answer is displayed. Having your
work in a notebook will allow you to compare your
answer to the correct answer.
Keep working problems until you get 4 out of 5 correct.
50. Way to Go!
You’ve finished reviewing
the notes for Adding and
Subtracting Polynomials.
Proceed to Module 9 Topic 1 Mastery
Assignment.