factoring trinomials using ac method
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This document provides a strategic intervention for factoring trinomials in Mathematics Grade 8. It tells the story of a boy named Nuj who dreams of becoming a successful teacher to support his family through education. When Nuj struggles with lessons, especially in math, he asks his brother Jim for help. The document aims to teach students how to factor trinomials by finding the greatest common factor.
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Linear cong slide 2
Here are the solutions to the given linear congruences:
2X≡ 1 (mod 17) has solution X=8
4x≡ 6(mod 18) has solution x=3
3x≡ 6(mod 18) has no solution since (3,18) does not divide 6
12x≡ 20(mod 28) has solution x=2
The system:
a) x≡1(mod 2), x≡1(mod 3) has the common solution x=1
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This document contains a lesson plan for teaching factoring non-perfect trinomials in Math 8. The lesson plan outlines intended learning outcomes, learning content including subject matter and reference materials, learning experiences through various activities, an evaluation, and assignment. Students will learn to define trinomials, factor non-perfect square trinomials, and apply factoring trinomials to geometric figures through guided practice with algebra tiles and examples.
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Factoring is writing a polynomial as a product of two or more polynomials. The main techniques for factoring polynomials are finding the greatest common factor, factoring trinomials of the form ax^2 + bx + c, using special factoring patterns like the difference and sum of squares, and factoring polynomials with four or more terms by grouping. The goal is to factor the polynomial completely into prime factors that cannot be further factored.
Linear cong slide 2
Here are the solutions to the given linear congruences:
2X≡ 1 (mod 17) has solution X=8
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3x≡ 6(mod 18) has no solution since (3,18) does not divide 6
12x≡ 20(mod 28) has solution x=2
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a) x≡1(mod 2), x≡1(mod 3) has the common solution x=1
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4. The remainder is 4.
5. The remainder is 7.
6. The remainder is 2.
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2) It describes how to factor trinomials of the form x2 + bx + c by finding two numbers whose sum is b and product is c.
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1) The distributive property, which involves multiplying each term of one polynomial with each term of the other.
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1. There are several methods for factoring polynomials outlined in the document: factoring using the distributive property, factoring the difference of two squares/cubes, factoring a perfect square trinomial, and factoring general trinomials using trial and error or grouping.
2. Factoring trinomials involves determining the signs in the factors based on the signs of the terms, then finding two factors of the constant term that satisfy the middle term.
3. More complex polynomials can be factored by grouping like terms or using special factoring patterns like the difference of squares/cubes.
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The document discusses exponential growth and decay functions. Exponential growth functions have a base greater than 1, modeling an increasing pattern from small to big numbers over time. Exponential decay functions have a base between 0 and 1, modeling a decreasing pattern from big to small numbers. Examples are provided of functions modeling exponential growth and decay, along with explanations of how to determine which type of function based on the base.
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This document discusses special products of binomials, including:
- (a + b)2 = a2 + 2ab + b2, known as a perfect-square trinomial
- (a - b)2 = a2 - 2ab + b2, also a perfect-square trinomial
- (a + b)(a - b) = a2 - b2, known as the difference of two squares
It provides examples of using these rules to simplify expressions involving binomials squared or multiplied together.
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Here are the remainders when dividing the given polynomials by the specified polynomials:
1. The remainder is 0. Therefore, x-1 is a factor of x3+3x2-4x+2.
2. The remainder is 5.
3. The remainder is 0. Therefore, x+2 is a factor of 2x3+5x2+3x+11.
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The document discusses factoring polynomials by finding the greatest common factor (GCF). It provides examples of factoring polynomials by finding the GCF of the numerical coefficients and variable terms. Students are then given practice problems to factor polynomials by finding the GCF and writing the factored form.
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This document provides instructions on factoring polynomials of various forms:
1) It explains how to factor polynomials by finding the greatest common factor (GCF).
2) It describes how to factor trinomials of the form x2 + bx + c by finding two numbers whose sum is b and product is c.
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The document discusses the Remainder Theorem and Factor Theorem. The Remainder Theorem states that if a polynomial p(x) is divided by a factor x - a, the remainder will be zero if x - a is a factor of p(x). The Factor Theorem is the reverse - if dividing a polynomial by x = a gives a zero remainder, then x - a is a factor of the polynomial. Both theorems relate the remainder of polynomial division to the factors of the polynomial.
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The document is about factoring polynomials, specifically factoring the sum and difference of cubes. It provides the formulas for factoring the sum and difference of cubes, along with examples of factoring expressions using those formulas. It also discusses factoring out the greatest common factor from polynomials.
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Solving multi step equations
The document contains examples of solving various types of algebraic equations including:
1) Equations with multiplication and subtraction or addition such as 2x - 4 = 8 and 5x + 10 = 80.
2) Equations with fractions such as 2/3x + 2 = 8.
3) Equations involving division such as x/5 + 2 = 8.
4) Equations with collecting like terms such as 4x + 6x + 20 = 80.
5) Equations using the distributive property such as 10x – 3x -12 = 4x – 9x + 48.
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This document provides instructions for solving systems of equations using elimination. It demonstrates eliminating variables by adding or subtracting equations. Sample systems are worked through, showing the steps of identifying which variable to eliminate, combining the equations accordingly, solving for one variable, then substituting back into the original equations to solve for the other. The solutions are checked in both equations to verify they satisfy the system.
Multiplying polynomials
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.
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This document provides an overview and activities on solving quadratic equations by factoring. It begins by defining quadratic equations and their standard form. Several activities are presented to practice identifying quadratic equations, rewriting them in standard form, and factoring trinomials of the form x^2 + bx + c. The final activity involves factoring quadratic equations to determine their roots. The document aims to build mastery of skills needed to solve quadratic equations using factoring techniques.
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This document provides instructions on how to evaluate functions by substituting values for variables and simplifying expressions using order of operations. It gives examples of evaluating various functions, such as f(x)=10+6x^2, by substituting the given value for x and simplifying. The key steps are to substitute the input value for the variable, then simplify the resulting expression using rules like PEMDAS. Floor and ceiling functions are also explained. Practice problems are provided to evaluate functions like f(x)=-5+3x for given x values.
Polynomials
Polynomials are algebraic expressions with whole number exponents. A polynomial can be classified based on the number of terms as a monomial, binomial, trinomial, or multinomial. It can also be classified based on degree, which is the highest sum of exponents of variables in any term. Polynomials can be added or subtracted by combining like terms, where like terms have the same variables and exponents. Unlike terms are kept separate when adding or subtracting polynomials.
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The student will learn to solve systems of equations by graphing. There are three types of solutions when graphing two equations: intersecting lines, which have a single solution where the lines cross; parallel lines, which have no solution; and coinciding lines, which have infinitely many solutions since the lines overlap completely. The student will practice graphing systems of equations and identifying the solution based on whether the lines intersect, are parallel, or coincide.
2.6 Linear Inequalities in Two Variables
This document discusses linear inequalities in two variables. It explains that linear inequalities look like linear equations but use <, >, ≥, or ≤ instead of =. Solutions to inequalities are ordered pairs that make the inequality true. Examples are given of checking solutions to inequalities and graphing various one- and two-variable inequalities on a coordinate plane.
8 - solving systems of linear equations by adding or subtracting
Here are the steps to solve this system of equations using elimination by multiplication:
1. Put the equations in standard form:
3x + y = 4
4x + 4y = 6
2. Determine which variable to eliminate. The y-terms have coefficients that are not the same, so multiply the top equation by 4 to make the y-coefficients the same:
12x + 4y = 16
4x + 4y = 6
3. Subtract the equations to eliminate y:
12x + 4y = 16
- (4x + 4y = 6)
8x = 10
x = 5/4
4. Plug x back into an original equation and
Linear inequalities in two variables
The document discusses linear inequalities in two variables and their graphical representations. It introduces the Cartesian coordinate system developed by Rene Descartes and its importance. It explains how to graph linear inequalities by first drawing the line as an equation, then determining whether to shade above or below the line based on whether a test point satisfies the inequality. Students are assigned to bring graphing paper, coloring materials, and a ruler to class on Monday to graph linear inequalities.
Factoring Perfect Square Trinomial - SIM
This document contains information and examples about perfect square trinomials in algebra. It defines a perfect square trinomial as having a first term and last term that are perfect squares, with a middle term that is twice the product of the square roots of the first and last terms. Examples are provided to illustrate how to identify if an expression is a perfect square trinomial or not. Several activity cards are included that provide practice identifying, factoring, and giving the factors of perfect square trinomial expressions.
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- 2. TITLE CARD Let’s Factor “T” 1
- 3. 2 There was a boy named Nuj, he dreamed that someday he will knock in the door of dreams and becomes a successful teacher. He is determined to reached his goals and elevates his family status in life and becomes an inspiration to others. To him he believes education is the key to his goals in life. “What the mind of man can conceive and believe, it can achieve.” —Napoleon Hill, 1883–1970, American speaker and motivational author
- 4. How will I factor this trinomials. I don’t know what is GCF? Brother can you help me please! Please! Brod, here’s how to do it, by the way what have you learned in your last lesson? Whenever Nuj is hard up in his lesson he ask his brother Jim to help him in his lessons specially in math . 3
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