1) By equating like terms, a=1 since the coefficients of x3 must be equal.
2) Equating the coefficients of x2 gives -3+b=5, so b=8.
3) Equating the coefficients of x gives -13+c=-8, so c=5.
05 perfect square, difference of two squaresmajapamaya
This document discusses perfect squares and the difference of two squares. It defines a perfect square as a number that can be expressed as a square, such as 9, 16, or 81. Any expression of the form a^2, (a+b)^2, or (k-h)^2 is also a perfect square. Perfect squares are helpful for expanding and factorizing expressions. For example, (c+d)^2 = c^2 + 2cd + d^2. The document also discusses how to find the area of a square when one side is increased by some amount b, using the formula (a+b)^2 = a^2 + 2ab + b^2. It concludes by explaining that
This document discusses factoring polynomials that are the difference of two squares using the formula a2 - b2 = (a + b)(a - b). It provides examples of factoring polynomials like x2 - 16, 9x2 - 100, and 36m2 - 49n4. It also lists 10 practice problems for factoring polynomials that are differences of two squares and references where readers can learn more.
Q1 week 1 (common monomial,sum & diffrence of two cubes,difference of tw...Walden Macabuhay
It consists of ten units in which the first unit focuses on the special products and factors. Its deals with the study of rational algebraic expressions. It aims to empower students with life – long learning and helps them to attain functional literacy. The call of the K to 12 curriculum allow the students to have an active involvement in learning through demonstration of skills, manifestations of communication skills, development of analytical and creative thinking and understanding of mathematical applications and connections.
The document discusses solving quadratic equations by factoring. It provides examples of factoring quadratic expressions into linear factors and setting each factor equal to 0 to solve for the roots. The key steps are: 1) write the equation in standard form, 2) factor completely, 3) set each factor equal to 0, 4) solve for the roots, and 5) check the solutions. It also provides examples of using this method to solve word problems involving quadratic equations.
2/27/12 Special Factoring - Sum & Difference of Two Cubesjennoga08
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.
Factoring 15.3 and 15.4 Grouping and Trial and Errorswartzje
This document discusses various methods for factoring trinomials of the form Ax^2 + Bx + C. It begins by outlining three main methods: trial and error, factoring by grouping, and the box method. It then provides examples of using the factoring by grouping method, demonstrating how to find two numbers whose product is AC and sum is B. The document also covers special cases like factoring the difference of squares using the form A^2 - B^2 = (A-B)(A+B), and factoring perfect square trinomials using the form A^2 + 2AB + B^2 = (A+B)^2. In all, it thoroughly explains the step-by
1. To factor non-perfect square trinomials, determine the factors of the leading term and the factors of the constant term that can be combined to form the middle term.
2. Trinomials of the form x^2 + bx + c can be factored into the product of two binomials by finding factor pairs of the constant term c that have a sum of b.
3. Examples are provided of factoring trinomials of the form a^2 + 10a + 25, m^2 + 10m + 21, and b^2 + 14b + 45.
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.
05 perfect square, difference of two squaresmajapamaya
This document discusses perfect squares and the difference of two squares. It defines a perfect square as a number that can be expressed as a square, such as 9, 16, or 81. Any expression of the form a^2, (a+b)^2, or (k-h)^2 is also a perfect square. Perfect squares are helpful for expanding and factorizing expressions. For example, (c+d)^2 = c^2 + 2cd + d^2. The document also discusses how to find the area of a square when one side is increased by some amount b, using the formula (a+b)^2 = a^2 + 2ab + b^2. It concludes by explaining that
This document discusses factoring polynomials that are the difference of two squares using the formula a2 - b2 = (a + b)(a - b). It provides examples of factoring polynomials like x2 - 16, 9x2 - 100, and 36m2 - 49n4. It also lists 10 practice problems for factoring polynomials that are differences of two squares and references where readers can learn more.
Q1 week 1 (common monomial,sum & diffrence of two cubes,difference of tw...Walden Macabuhay
It consists of ten units in which the first unit focuses on the special products and factors. Its deals with the study of rational algebraic expressions. It aims to empower students with life – long learning and helps them to attain functional literacy. The call of the K to 12 curriculum allow the students to have an active involvement in learning through demonstration of skills, manifestations of communication skills, development of analytical and creative thinking and understanding of mathematical applications and connections.
The document discusses solving quadratic equations by factoring. It provides examples of factoring quadratic expressions into linear factors and setting each factor equal to 0 to solve for the roots. The key steps are: 1) write the equation in standard form, 2) factor completely, 3) set each factor equal to 0, 4) solve for the roots, and 5) check the solutions. It also provides examples of using this method to solve word problems involving quadratic equations.
2/27/12 Special Factoring - Sum & Difference of Two Cubesjennoga08
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.
Factoring 15.3 and 15.4 Grouping and Trial and Errorswartzje
This document discusses various methods for factoring trinomials of the form Ax^2 + Bx + C. It begins by outlining three main methods: trial and error, factoring by grouping, and the box method. It then provides examples of using the factoring by grouping method, demonstrating how to find two numbers whose product is AC and sum is B. The document also covers special cases like factoring the difference of squares using the form A^2 - B^2 = (A-B)(A+B), and factoring perfect square trinomials using the form A^2 + 2AB + B^2 = (A+B)^2. In all, it thoroughly explains the step-by
1. To factor non-perfect square trinomials, determine the factors of the leading term and the factors of the constant term that can be combined to form the middle term.
2. Trinomials of the form x^2 + bx + c can be factored into the product of two binomials by finding factor pairs of the constant term c that have a sum of b.
3. Examples are provided of factoring trinomials of the form a^2 + 10a + 25, m^2 + 10m + 21, and b^2 + 14b + 45.
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.
Algebra Presentation on Topic Modulus Function and PolynomialsAnzaLubis
Algebra Presentation on Topic Modulus Function and Polynomials
Created by: Anza Sahel Lubis and Jonathan Briant Wijaya 12 A
Performance Task Binus School Bekasi
The document discusses perfect square trinomials and how to factor them. It provides examples of factoring various square trinomials using the properties that the first and last terms must be perfect squares, and the middle term must be twice the product of the square roots of the first and last terms. It then has students practice factoring several square trinomial examples on their own.
This document discusses factoring perfect-square trinomials, which are expressions of the form a^2x^2 ± 2abx + b^2. It provides examples of factoring different trinomials like x^2 - 49, 9a^2 - 25b^2, and 16x^4 - 64y^2. Readers are instructed to practice factoring additional examples like x^2 - 49, c^2 - 81, 4x^2 - 25, a^2b^2 - 64c^2, and 25x^2 - 121y^2.
This document discusses factoring the sum and difference of two cubes. It explains that the sum or difference of two cubes can be factored into a binomial times a trinomial, with the first term of the trinomial being the cube root of the first term, the second term being the product of the cube roots, and the third term being the cube root of the second term. It provides an example of factoring 27x3 - 125 to show the process.
To factor perfect square trinomials:
1. Get the square root of the first and last terms.
2. List down the square root as the sum or difference of two terms.
3. Factor the expression into the form (term 1 ± term 2)2.
The document provides examples of factoring quadratic trinomials. It gives the steps to factor expressions such as 2x^2 - x - 6, 10x^2 + 3x - 1, and 3x^2 - x - 2. The steps involve multiplying the coefficient of x^2 by the constant term, finding the factors of the resulting number that sum to the coefficient of x, and then grouping the expression and factoring using the common factor.
This document provides guidance on identifying and factoring perfect square trinomials in algebra. It begins with a definition of a perfect square trinomial as a trinomial that results from squaring a binomial. Examples are provided to illustrate this. Several activity cards are then presented to help students practice determining if an expression is a perfect square trinomial, completing the terms, factoring, and more. An enrichment card adds an assessment with multiple choice questions. Key steps for factoring a perfect square trinomial are outlined, such as taking the square root of the first and last terms. An answer card provides the solutions to the activities. Sources are listed at the end.
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.
The document provides instructions for factoring quadratic trinomials using 4 examples. It explains that you write the trinomial as two parentheses, factor the constant term into the parentheses, then check that the factors give the middle term of the original expression when multiplied out. The process involves 4 steps and is demonstrated factoring expressions like x^2 + 10x + 24 and x^2 - 8x + 15.
This document provides instruction on perfect square trinomials including defining them, identifying them, factoring them, and working practice problems. It begins by defining a perfect square trinomial as the result of squaring a binomial with the first and last terms being perfect squares and the middle term being twice the product of the square roots of the first and last terms. Examples are provided to illustrate. The document then provides guidance, activities, and an assessment to practice identifying, factoring, and working with perfect square trinomials.
This document provides examples and practice problems for algebraic expressions involving products, quotients, and simplification of expressions with variables. It covers finding products and quotients of expressions with variables. It also covers simplifying expressions by combining like terms, distributing operations, and factoring. The document aims to help students master skills in working with algebraic expressions.
This document discusses quadratic equations. It defines a quadratic equation as having the standard form ax^2 + bx + c, where a, b, and c are constants and x is the variable. It describes three methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula. Factoring works when the equation can be rewritten as a product of two binomials. Completing the square is used when factoring is not possible. It involves rearranging terms to create a perfect square trinomial. The quadratic formula can be used to solve any quadratic equation by substituting the a, b, and c values.
Solving quadratic equation using completing the squareMartinGeraldine
1. The document provides steps for solving quadratic equations using completing the square. It involves isolating terms with x, finding the square of half the coefficient of x, completing the square of the left side of the equation, applying the square root property, and solving for the roots.
2. Three examples are shown applying these steps: finding the solution set of x^2 - 18x = -17 to be x = 17 and x = 1; solving x^2 + 10x - 11 = 0 to get x = 1 and x = -11; and solving x^2 = -24 + 10x to get x = 6 and x = 4.
This will help you in factoring sum and difference of two cubes.
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The document discusses the zero factor theorem and provides examples of using it to solve quadratic equations. The zero factor theorem states that if p and q are algebraic expressions, then pq = 0 if and only if p = 0 or q = 0. This means a quadratic equation can be solved by factoring it into two linear factors and setting each factor equal to zero. Five examples are provided that show factoring quadratic equations, applying the zero factor theorem to set the factors equal to zero, and solving for the roots.
This document provides examples for rewriting linear equations between the slope-intercept form (y=mx+b) and standard form (Ax + By = C).
It begins with examples of rewriting equations from standard form to slope-intercept form and identifying the slope (m) and y-intercept (b). Then it provides examples of rewriting from slope-intercept form to standard form. Finally, it provides a series of practice problems for rewriting linear equations between the two forms.
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Botany Downs Secondary College in Auckland, New Zealand has been selected as part of the Microsoft Innovative Schools program for its innovative approaches in three areas: physical learning environments that encourage collaboration, personalized learning environments using data to tailor instruction, and professional learning communities among teachers. The school features open classroom designs and technology access to support individualized learning. Student progress is closely tracked to help teachers personalize instruction. Professional development emphasizes collaborative inquiry among teachers.
Giftblooms presents top valentine day gift ideas for him. It has flowers,chocolates,cake,stuffed animals and many more for the valentine gifts delivery
El documento describe diferentes teorías de la motivación, incluyendo la jerarquía de necesidades de Maslow y las teorías X y Y de McGregor. Según Maslow, los seres humanos tienen una jerarquía piramidal de necesidades que van desde las fisiológicas en la base hasta la autorrealización en la cima. Las teorías X y Y de McGregor proponen dos estilos de dirección: la teoría X asume que los empleados son perezosos mientras que la teoría Y supone que los empleados buscan responsabilidades
Algebra Presentation on Topic Modulus Function and PolynomialsAnzaLubis
Algebra Presentation on Topic Modulus Function and Polynomials
Created by: Anza Sahel Lubis and Jonathan Briant Wijaya 12 A
Performance Task Binus School Bekasi
The document discusses perfect square trinomials and how to factor them. It provides examples of factoring various square trinomials using the properties that the first and last terms must be perfect squares, and the middle term must be twice the product of the square roots of the first and last terms. It then has students practice factoring several square trinomial examples on their own.
This document discusses factoring perfect-square trinomials, which are expressions of the form a^2x^2 ± 2abx + b^2. It provides examples of factoring different trinomials like x^2 - 49, 9a^2 - 25b^2, and 16x^4 - 64y^2. Readers are instructed to practice factoring additional examples like x^2 - 49, c^2 - 81, 4x^2 - 25, a^2b^2 - 64c^2, and 25x^2 - 121y^2.
This document discusses factoring the sum and difference of two cubes. It explains that the sum or difference of two cubes can be factored into a binomial times a trinomial, with the first term of the trinomial being the cube root of the first term, the second term being the product of the cube roots, and the third term being the cube root of the second term. It provides an example of factoring 27x3 - 125 to show the process.
To factor perfect square trinomials:
1. Get the square root of the first and last terms.
2. List down the square root as the sum or difference of two terms.
3. Factor the expression into the form (term 1 ± term 2)2.
The document provides examples of factoring quadratic trinomials. It gives the steps to factor expressions such as 2x^2 - x - 6, 10x^2 + 3x - 1, and 3x^2 - x - 2. The steps involve multiplying the coefficient of x^2 by the constant term, finding the factors of the resulting number that sum to the coefficient of x, and then grouping the expression and factoring using the common factor.
This document provides guidance on identifying and factoring perfect square trinomials in algebra. It begins with a definition of a perfect square trinomial as a trinomial that results from squaring a binomial. Examples are provided to illustrate this. Several activity cards are then presented to help students practice determining if an expression is a perfect square trinomial, completing the terms, factoring, and more. An enrichment card adds an assessment with multiple choice questions. Key steps for factoring a perfect square trinomial are outlined, such as taking the square root of the first and last terms. An answer card provides the solutions to the activities. Sources are listed at the end.
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.
The document provides instructions for factoring quadratic trinomials using 4 examples. It explains that you write the trinomial as two parentheses, factor the constant term into the parentheses, then check that the factors give the middle term of the original expression when multiplied out. The process involves 4 steps and is demonstrated factoring expressions like x^2 + 10x + 24 and x^2 - 8x + 15.
This document provides instruction on perfect square trinomials including defining them, identifying them, factoring them, and working practice problems. It begins by defining a perfect square trinomial as the result of squaring a binomial with the first and last terms being perfect squares and the middle term being twice the product of the square roots of the first and last terms. Examples are provided to illustrate. The document then provides guidance, activities, and an assessment to practice identifying, factoring, and working with perfect square trinomials.
This document provides examples and practice problems for algebraic expressions involving products, quotients, and simplification of expressions with variables. It covers finding products and quotients of expressions with variables. It also covers simplifying expressions by combining like terms, distributing operations, and factoring. The document aims to help students master skills in working with algebraic expressions.
This document discusses quadratic equations. It defines a quadratic equation as having the standard form ax^2 + bx + c, where a, b, and c are constants and x is the variable. It describes three methods for solving quadratic equations: factoring, completing the square, and using the quadratic formula. Factoring works when the equation can be rewritten as a product of two binomials. Completing the square is used when factoring is not possible. It involves rearranging terms to create a perfect square trinomial. The quadratic formula can be used to solve any quadratic equation by substituting the a, b, and c values.
Solving quadratic equation using completing the squareMartinGeraldine
1. The document provides steps for solving quadratic equations using completing the square. It involves isolating terms with x, finding the square of half the coefficient of x, completing the square of the left side of the equation, applying the square root property, and solving for the roots.
2. Three examples are shown applying these steps: finding the solution set of x^2 - 18x = -17 to be x = 17 and x = 1; solving x^2 + 10x - 11 = 0 to get x = 1 and x = -11; and solving x^2 = -24 + 10x to get x = 6 and x = 4.
This will help you in factoring sum and difference of two cubes.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
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https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
The document discusses the zero factor theorem and provides examples of using it to solve quadratic equations. The zero factor theorem states that if p and q are algebraic expressions, then pq = 0 if and only if p = 0 or q = 0. This means a quadratic equation can be solved by factoring it into two linear factors and setting each factor equal to zero. Five examples are provided that show factoring quadratic equations, applying the zero factor theorem to set the factors equal to zero, and solving for the roots.
This document provides examples for rewriting linear equations between the slope-intercept form (y=mx+b) and standard form (Ax + By = C).
It begins with examples of rewriting equations from standard form to slope-intercept form and identifying the slope (m) and y-intercept (b). Then it provides examples of rewriting from slope-intercept form to standard form. Finally, it provides a series of practice problems for rewriting linear equations between the two forms.
For more instructional resources, CLICK me here!
https://tinyurl.com/y9muob6q
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https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Botany Downs Secondary College in Auckland, New Zealand has been selected as part of the Microsoft Innovative Schools program for its innovative approaches in three areas: physical learning environments that encourage collaboration, personalized learning environments using data to tailor instruction, and professional learning communities among teachers. The school features open classroom designs and technology access to support individualized learning. Student progress is closely tracked to help teachers personalize instruction. Professional development emphasizes collaborative inquiry among teachers.
Giftblooms presents top valentine day gift ideas for him. It has flowers,chocolates,cake,stuffed animals and many more for the valentine gifts delivery
El documento describe diferentes teorías de la motivación, incluyendo la jerarquía de necesidades de Maslow y las teorías X y Y de McGregor. Según Maslow, los seres humanos tienen una jerarquía piramidal de necesidades que van desde las fisiológicas en la base hasta la autorrealización en la cima. Las teorías X y Y de McGregor proponen dos estilos de dirección: la teoría X asume que los empleados son perezosos mientras que la teoría Y supone que los empleados buscan responsabilidades
The Chariot Card summarizes as follows:
[1] The Chariot Card represents continuous movement, a new journey, and victory.
[2] It indicates that a new professional challenge will soon be available that will lead to success.
[3] The figure on the card, representing the person, will experience complete control of their destiny and elements of success.
The Star Card summarizes as follows:
[1] The Star Card reflects the relationship developing between the person and what they are currently experiencing.
[2] It indicates the beginning of a new beneficial cycle in love life and a powerful force of attraction between two people.
[3] The card overall represents love
The National Women Business Owners Corporation has certified Claridge Products as a Woman Business Enterprise. Claridge Products manufactures dry erase markerboards and visual display products in a 500,000 square foot facility and has a reputation for quality, service, and innovation over 60 years. Claridge offers customization, quick shipping, and knowledgeable support staff to suit customers' needs.
El documento habla sobre la profesión de submarinista. Un submarinista es una persona que explora y trabaja bajo el agua, ya sea en buceo recreativo o en trabajos profesionales como la inspección y reparación de estructuras subacuáticas. Se requiere de entrenamiento y certificaciones especiales para trabajar de forma segura bajo el agua, ya que las condiciones son más desafiantes que en la superficie.
Buddhist psychology uses meditation to develop concentration and insight. Through focusing on the breath, negative thoughts and repressed emotions surface without judgment. Deeper concentration is achieved by feeling warmth and observing thoughts without attachment. Insight is generated by exploring the origins of negative thoughts and emotions, which alters the thoughts and offsets the negative feelings, resulting in enlightenment.
Brigada de Cultura Ambiental y Turística BRICATURLaura Domínguez
La Brigada de Cultura Ambiental y Turística es un grupo que trabaja para promover el desarrollo sostenible del turismo en Venezuela a través de acciones de sensibilización ambiental y turística dirigidas a la comunidad y los turistas. Su misión es aumentar la valoración y el uso racional del patrimonio natural y cultural a través de capacitación y proyectos propios o en colaboración con otras organizaciones. Los miembros de la brigada, que son estudiantes universitarios, se responsabilizan de las comunidades cercanas y
AC/DC is a hard rock band formed in 1973 in Sydney, Australia by brothers Malcolm and Angus Young. They are considered legendary for their iconic rock songs over four decades. The band currently consists of five members - Brian Johnson, Angus Young, Stevie Young, Cliff Williams, and Chris Slade. Some of their most popular songs include "It's a Long Way to the Top", "Highway to Hell", and "Thunderstruck".
Stars are born from clouds of gas and dust called nebulae. Over billions of years, stars progress through various stages as they age. Lower mass stars begin as protostars and become main sequence stars fueled by nuclear fusion. As their hydrogen runs out, they become red giants and eventually white dwarfs. Higher mass stars explode as supernovae at the end of their lives, leaving behind neutron stars or black holes.
Oportunidades de financiación de la I+D+i. CDTI. Eduardo Cotillas. CDTI.CTAEX
El documento describe las diferentes oportunidades de financiación para proyectos de I+D e innovación tecnológica que ofrece el Centro para el Desarrollo Tecnológico Industrial (CDTI) para empresas del sector agroalimentario, incluyendo proyectos nacionales e internacionales, programas multilaterales y bilaterales, y una red exterior para promover la transferencia de tecnología.
Lessons From The Dell Brand Newsroom, Digiday Content Marketing Summit, Augus...Digiday
The document discusses lessons learned from Dell's brand newsroom. It emphasizes putting customers at the center and championing their success. Key lessons are to stay agile and responsive by acting on data insights, understand the brand is the publisher but doesn't have all the rights of traditional media, and maintain a customer-centric focus by portraying customers as the heroes in their own stories and acting as their champion rather than savior.
Connection is an IT staffing company that helps organizations address challenges in filling open IT positions. It has decades of experience placing IT professionals in both short-term contract and permanent roles. Connection can source candidates more broadly than internal HR departments and offers flexible contract-to-hire options. Its services help organizations fill positions faster while minimizing costs and risks.
The document provides instructions for graphing and solving various types of quadratic equations. It defines standard form, vertex form, and intercept form of quadratics. It explains how to graph quadratics by finding the vertex and intercepts. Methods covered include factoring, taking square roots, completing the square, and using the quadratic formula. Examples are included to demonstrate each process.
Partial Fraction ppt presentation of engg mathjaidevpaulmca
The document discusses algebraic fractions and partial fractions. It begins by defining algebraic fractions as fractions where the numerator and denominator are polynomial expressions. It then provides examples of proper and improper algebraic fractions. The key points are that a fraction is proper if the numerator's degree is less than the denominator's, and improper otherwise. The document goes on to explain how to express improper and proper fractions as a sum of partial fractions by setting up equations and using special values of x to solve for the coefficients.
The document contains a 15 question multiple choice test on polynomials. It covers topics like finding the number of zeros of a polynomial from its graph, identifying polynomials based on their zeros, finding the sum and product of the zeros of a quadratic polynomial, and determining the remaining zeros if one zero is given. It also includes word problems involving dividing polynomials and finding zeros. The test is meant to assess students' conceptual understanding of polynomials and their ability to solve subjective questions similar to those in board exams.
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -1.
Aieee 2003 maths solved paper by fiitjeeMr_KevinShah
1. The function f maps natural numbers to integers such that even numbers map to themselves divided by 2 and odd numbers map to themselves minus 1. This function is one-to-one but not onto.
2. If two roots of a quadratic equation form an equilateral triangle with the origin, then the coefficients a and b satisfy the relationship a^2 = 3b.
3. If the modulus of the product of two non-zero complex numbers z and ω is 1, and the difference of their arguments is 2π, then their product ωz is equal to -i.
This document contains 60 multiple choice questions from a past mathematics exam. The questions cover a range of topics including relations, functions, complex numbers, matrices, determinants, quadratic equations, arithmetic and geometric progressions, binomial expansions, trigonometry, calculus, differential equations, vectors, conic sections, and three-dimensional geometry. For each question, four choices are given and the student must select the correct answer.
Linear equation in one variable for class VIII by G R Ahmed MD. G R Ahmed
This document discusses linear equations in one variable. It defines linear equations as those where the highest power of the variable is 1. It provides examples of linear and non-linear equations. It also discusses how to determine if a value is a solution to a linear equation and how to solve linear equations using the addition, subtraction, and multiplication properties of equality. It provides examples of solving linear equations with fractions and with consecutive integers.
1) The document explains various methods for dividing and factoring polynomials, including: dividing polynomials using long division; using Ruffini's rule to divide polynomials; applying the remainder theorem and factor theorem; and factoring polynomials through finding common factors, using identities, solving quadratic equations, and finding polynomial roots.
2) Specific factorization methods covered are removing common factors, using identities like a^2 - b^2, factoring quadratic trinomials, using the remainder theorem and Ruffini's rule to find factors for polynomials of degree greater than two, and identifying irreducible polynomials.
3) Additional algebraic identities explained are for cubing binomials like (a ± b)^3 and taking the square of trinomial
This document provides notes on functions and quadratic equations from Additional Mathematics Form 4. It includes:
1) Definitions of functions, including function notation f(x) and the relationship between objects and images.
2) Methods for solving quadratic equations, including factorisation, completing the square, and the quadratic formula.
3) Properties of quadratic functions like finding the maximum/minimum value and sketching the graph.
4) Solving simultaneous equations involving one linear and one non-linear equation through substitution.
5) Conversions between index and logarithmic forms and basic logarithm laws.
This document provides notes on key concepts in additional mathematics including:
1) Functions such as f(x) = x + 3 and finding the object and image of a function.
2) Solving quadratic equations using factorisation and the quadratic formula. Types of roots are discussed.
3) Sketching quadratic functions by finding the y-intercept, maximum/minimum values, and a third point. Quadratic inequalities are also covered.
4) Methods for solving simultaneous equations including substitution when one equation is nonlinear.
5) Properties of exponents and logarithms, and how to solve exponential and logarithmic equations.
The document provides examples of factoring quadratic expressions using different techniques like finding the greatest common factor, difference of squares, and perfect square trinomials. It also reviews the formulas for factoring the square of a sum and square of a difference and provides a test review with additional practice problems. The review is intended to prepare students to factor a variety of quadratic expressions using various factoring methods.
This document summarizes three methods for solving systems of linear equations: graphing, substitution, and elimination. It provides examples of solving systems of two equations using each method. Graphing involves plotting the lines defined by each equation on a coordinate plane and finding their point of intersection. Substitution involves isolating a variable in one equation and substituting it into the other equation. Elimination involves adding or subtracting multiples of equations to remove a variable and solve for the remaining variable.
1. The height of the hill is equal to 1/2tanα, where α is the angle of elevation of the top of the hill from each of the vertices A, B, and C of a horizontal triangle.
2. The solutions of the equation 4cos^2x + 6sin^2x = 5 are x = π/4 + nπ/2, where n is any integer.
3. A four-digit number formed using the digits 1, 2, 3, 4, 5 without repetition has a 1/3 probability of being divisible by 3.
This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
1. A function f maps objects to images. To find the inverse function f-1, change f(x) to y and solve for x in terms of y.
2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solve
The document contains 50 multiple choice questions covering various topics in mathematics including functions, trigonometry, calculus, probability, matrices and linear algebra. The questions test concepts such as one-to-one functions, inverse trigonometric functions, limits, derivatives, integrals, probability distributions, matrices, and linear transformations.
The document contains 50 multiple choice questions covering various topics in mathematics including functions, trigonometry, calculus, probability, matrices and linear algebra. The questions test concepts such as one-to-one functions, inverse trigonometric functions, limits, derivatives, integrals, probability, matrices, vectors, and linear transformations.
The document contains 6 problems related to algebra and numbers along with their solutions. Problem 1 involves a number guessing game between two players and determining the minimum number of rounds needed. Problem 2 examines properties of a polynomial where the polynomial equals certain values for distinct integer inputs. Problem 3 finds all integer solutions to a system of equations involving cubes of variables. Problem 4 determines the value of a polynomial of degree 8 at a particular input, given its values at other integers. Problems 5 and 6 involve finding the smallest integer greater than an expression and the minimum possible value of a product of variables, respectively, given an equation relating the variables.
1. A differential equation is an equation that relates an unknown function with some of its derivatives. The document provides a step-by-step example of solving a differential equation to find the xy-equation of a curve with a given gradient condition.
2. The key steps are: (1) write the derivative term as a fraction, (2) integrate both sides, (3) apply the initial condition to determine the constant term, (4) write the final function relationship.
3. Common types of differential equations discussed are separable first order equations, where the derivative terms can be isolated by dividing both sides.
This document contains 3 problems:
1) Showing that a sequence is increasing and less than 3.
2) Calculating the average value of a function over an interval.
3) Finding the solution to a partial differential equation of the form 2xyy' – y^2 + x^2 = 0. The solution is found to be y = √cx - x^2.
2. Because this is an equation both sides have to be equal Since the x3 gotten in a(x-1)3 =a (x3-3x2+3x-1) is the only term of that degree on the right side of the equation a=1 because the X3*1 = X3. 1(x3-3x2+3x-1) = (x3-3x2+3x-1) b(x-1)2= b(x2-2x+1) All the x2 term’s coefficients on the left side must add up to 5 that -3+b = 5 allows you to solve for b. b=8 Compute the value of c such that x3+5x2-8x+3= a(x-1)3+b(x-1)2+c(x-1)+d
3. x3-3x2+3x-1+b(x2-2x+1) After finding b plug it back x3-3x2+3x-1+8(x2-2x+1) into the equation. x3-3x2+3x-1 + 8x2-16x+8 x3+5x2 -13x+7 -13+c=-8 Using the x term values from the c=5 othertwo terms and -8 you can solve for c. Compute the value of c such that x3+5x2-8x+3= a(x-1)3+b(x-1)2+c(x-1)+d