Factor theorem solving cubic equations
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The document discusses using the factor theorem to find factors of polynomials and solve cubic equations. The factor theorem states that if a polynomial P(x) is divisible by (x - a), then P(a) must be equal to 0. It can be used to test if a linear expression is a factor and to find the factors of a polynomial equation. Cubic equations can be solved by applying the factor theorem to find a linear factor, then reducing the cubic to a quadratic equation. Examples are provided to demonstrate solving cubic equations using this process.
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Factor theorem solving cubic equations
The document discusses using the factor theorem to find factors of polynomials and solve cubic equations. The factor theorem states that if a polynomial P(x) is divisible by (x - a), then P(a) must be equal to 0. It can be used to test if a linear expression is a factor and to find the factors of a polynomial equation. Cubic equations can be solved by applying the factor theorem to find a linear factor, then reducing the cubic to a quadratic equation. Examples are provided to demonstrate solving cubic equations using this process.
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This document discusses the remainder theorem for polynomials. The remainder theorem states that if a polynomial P(x) is divided by a linear divisor (x - a), the remainder is equal to P(a). The document provides examples of using the remainder theorem to find the remainder when dividing polynomials by linear expressions like (x + 1) or (x - a). It explains that the remainder theorem provides a simpler method than long division for calculating the remainder.
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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.
Factor theorem
This document discusses the remainder theorem for polynomials. The remainder theorem states that if a polynomial P(x) is divided by a linear divisor (x - a), the remainder is equal to P(a). The document provides examples of using the remainder theorem to find the remainder when dividing polynomials by linear expressions like (x + 1) or (x - a). It also discusses using the theorem to solve for unknown values in polynomials.
Remainder theorem and factorization of polynomials
The remainder theorem states that the remainder of dividing a polynomial P(x) by (x-a) is equal to P(a). Some examples of using the remainder theorem to find the remainder of polynomial divisions are worked out. The document also discusses that a number a is a root of a polynomial P(x) if P(a) equals 0. If a is a root, then (x-a) is a factor of the polynomial P(x). Examples of finding roots and corresponding factors of polynomials are provided.
Remainder theorem
The document discusses the remainder theorem for polynomials. It defines the division algorithm for polynomials which divides a polynomial P(x) by (x-c) to get a unique quotient polynomial Q(x) and remainder R. The remainder theorem then states that the remainder R is equal to the value of P(c). The document proves the theorem and provides examples of using it to find the remainder when one polynomial is divided by another. It also provides exercises for students to find remainders using the theorem.
2 2 synthetic division, remainder & factor theorems
1. The document discusses synthetic division and the Remainder and Factor Theorems. It provides examples of using synthetic division to find the remainder when dividing polynomials and to check if an expression is a factor.
2. Readers are prompted to try dividing polynomials using synthetic division and to determine if expressions are factors of given polynomials.
3. An extra example shows using synthetic division to find the remaining roots of a polynomial equation after one root is identified.
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This document discusses polynomials and partial fractions. It covers arithmetic operations on polynomials, identities involving polynomials, and dividing polynomials. Specifically, it teaches how to:
1) Perform addition, subtraction, and multiplication on polynomial expressions.
2) Identify polynomial equations and identities based on their properties.
3) Use the long division process to divide one polynomial by another.
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The document is a presentation on polynomials. It defines a polynomial as an expression that can contain constants, variables, and exponents, but cannot contain division by a variable. It discusses the key characteristics of polynomials including their degree, standard form, zeros, factoring, and algebraic identities. Examples are provided to illustrate different types of polynomials like monomials, binomials, trinomials, and how to add, subtract, multiply and divide polynomials.
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Let's think step by step:
1) For x2 + 2x + 5 to be a factor of x4 + px2 + q, when we divide x4 + px2 + q by x2 + 2x + 5, the remainder should be 0.
2) Dividing x4 + px2 + q by x2 + 2x + 5, the quotient will be x2 and the remainder will be p - 5.
3) For the remainder to be 0, we must have p - 5 = 0.
Therefore, p = 5
4) Putting p = 5 in the original expression, we get:
x4 + 5x2 + q
5) Comparing the constant
Zeros of p(x)
This document discusses various methods for finding the zeros or roots of polynomial functions, including factoring, factor theorem, synthetic division, and using the principle that every polynomial of degree n has n zeros. It provides examples of finding the zeros of polynomials by factorization, using a given zero to find other zeros through synthetic division, and identifying which numbers are zeros of various polynomials. Exercises are included for students to practice finding remaining zeros given one zero and identifying polynomial factors.
Maths project abhi.pptx [autosaved]
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i. The document discusses polynomials, including definitions, types of polynomials based on degree, and key concepts like factorization, the relationship between zeros and coefficients of quadratic and cubic polynomials, and graphs of polynomials.
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iii. Examples of polynomial factorization and the relationship between zeros and coefficients are provided.
Shubhanshu math project work , polynomial
This document summarizes key concepts about polynomials:
1) It defines a polynomial as an algebraic expression involving variables and their powers, and defines the degree of a polynomial as the highest power of the variable.
2) It describes different types of polynomials based on degree, including constant, linear, quadratic, cubic, and biquadratic polynomials.
3) It explains that a zero of a polynomial is a value that makes the polynomial equal to zero when substituted for the variable, and discusses the relationship between zeros and coefficients of polynomials.
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1) The document describes performing synthetic division to divide a polynomial by a linear term (x - a).
2) It works through an example where the divisor is (x + 3), finding that a = -3.
3) It then sets up the synthetic division algorithm and carries out the steps, obtaining the quotient polynomial (3x^2 - 13x + 42) and remainder -121.
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This document provides instructions for using synthetic division to divide polynomials. It contains the following key points:
1. Synthetic division can be used to divide polynomials when the divisor has a leading coefficient of 1 and there is a coefficient for every power of the variable in the numerator.
2. The procedure involves writing the terms of the numerator in descending order, bringing down the constant of the divisor, multiplying and adding down the columns to obtain the coefficients of the quotient polynomial and the remainder.
3. An example problem walks through each step of synthetic division to divide (5x^4 - 4x^2 + x + 6) / (x - 3), obtaining a quotient of 5x^3 + 15
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This document discusses types of polynomials including constant, linear, quadratic, cubic, and bi-quadratic polynomials. It defines a zero of a polynomial as a real number where the polynomial equals 0. Geometrically, linear polynomials intersect the x-axis at one point, quadratic polynomials form parabolas that can open up or down, and cubic polynomials can have up to three zeros where they intersect the x-axis. Polynomials are used in engineering, economics, physics, and industry to model and describe real-world phenomena like roller coaster curves, price variations over time, energy and voltage differences.
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Let's think step by step:
1) For x2 + 2x + 5 to be a factor of x4 + px2 + q, when we divide x4 + px2 + q by x2 + 2x + 5, the remainder should be 0.
2) Dividing x4 + px2 + q by x2 + 2x + 5, the quotient will be x2 and the remainder will be p - 5.
3) For the remainder to be 0, we must have p - 5 = 0.
Therefore, p = 5
4) Putting p = 5 in the original expression, we get:
x4 + 5x2 + q
5) Comparing the constant
Zeros of p(x)
This document discusses various methods for finding the zeros or roots of polynomial functions, including factoring, factor theorem, synthetic division, and using the principle that every polynomial of degree n has n zeros. It provides examples of finding the zeros of polynomials by factorization, using a given zero to find other zeros through synthetic division, and identifying which numbers are zeros of various polynomials. Exercises are included for students to practice finding remaining zeros given one zero and identifying polynomial factors.
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The document discusses polynomials and their properties. It defines polynomials and different types based on degree. It describes how to find the zeroes of a polynomial by factorizing and setting factors equal to 0. The relationship between the coefficients and zeroes of a polynomial is explained. Specifically, the sum of the zeroes equals the negative of the coefficient of the highest-degree term divided by the leading coefficient, and the product of the zeroes equals the coefficient of the constant term divided by the leading coefficient. The division algorithm for polynomials is also introduced.
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The remainder theorem states that if a polynomial p(x) is divided by the linear polynomial x - a, the remainder is equal to p(a). The proof involves writing p(x) as the product of (x - a) and a quotient polynomial q(x), plus a remainder r(x). Since the degree of r(x) must be less than the degree of x - a, which is 1, r(x) must be a constant. Therefore, the remainder is equal to the value of p(x) when x is substituted with a.
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The document also discusses using the interval test and graphing polynomials to determine where a polynomial is positive or negative. It provides examples of sketching graphs of polynomials based on their degree and characteristics at roots, such as whether the graph will cross or bounce at roots depending on if the root's multiplicity is odd or even.
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Shubhanshu math project work , polynomial
This document summarizes key concepts about polynomials:
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2) It describes different types of polynomials based on degree, including constant, linear, quadratic, cubic, and biquadratic polynomials.
3) It explains that a zero of a polynomial is a value that makes the polynomial equal to zero when substituted for the variable, and discusses the relationship between zeros and coefficients of polynomials.
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Polynomials are mathematical expressions constructed from variables and constants using addition, subtraction, multiplication, and exponents of whole numbers. They appear in many areas of mathematics and science. Polynomials can be used to form equations that model problems in various domains. They also define polynomial functions that are used in fields like physics, chemistry, economics, and social sciences. Polynomials are classified based on their degree, with linear polynomials having degree 1 and quadratic polynomials degree 2. The maximum number of zeroes a polynomial can have is equal to its degree.
5.5 Zeros of Polynomial Functions
* Evaluate a polynomial using the Remainder Theorem.
* Use the Factor Theorem to solve a polynomial equation.
* Use the Rational Zero Theorem to find rational zeros.
* Find zeros of a polynomial function.
* Use the Linear Factorization Theorem to find polynomials with given zeros.
* Use Descartes’ Rule of Signs.
Cl 9 Chapter 2.ppt
This document discusses polynomials. It defines a polynomial as an expression constructed from variables and constants using addition, subtraction, multiplication, and exponents. Polynomials appear in many areas of mathematics and science. The document then discusses the terms, coefficients, and different types of polynomials including linear, quadratic, cubic, and constant polynomials. It also discusses the degree and maximum number of zeroes of each type of polynomial. Finally, it discusses using the division algorithm to find quotients, remainders, and factors of polynomials.
Synthetic and Remainder Theorem of Polynomials.ppt
This document discusses synthetic division, a shortcut method for long division of polynomials when dividing by divisors of the form x - k. Synthetic division works by setting up an array and adding terms in columns, multiplying the results by -k. The remainder obtained has an important interpretation related to the Remainder Theorem - evaluating the polynomial function at x = k. The Factor Theorem also relates dividing a polynomial by (x - k) to determining if (x - k) is a factor. Examples demonstrate using synthetic division and these theorems to factor polynomials and find zeros.
Advance algebra
This document provides information about several advanced algebra topics:
1. The Remainder Theorem states that when dividing a polynomial f(x) by x-c, the remainder is equal to f(c).
2. The Factor Theorem states that if f(c) = 0, then x-c is a factor of the polynomial.
3. Synthetic division is a method for dividing polynomials without subtracting, by multiplying the coefficients.
4. The Rational Zeros Theorem can be used to find all possible rational zeros of a polynomial by considering factors of the leading coefficient and constant term.
Chapter 3
The document discusses various numerical methods for finding the roots or zeros of equations, including closed and open methods. Closed methods like bisection and false position trap the root within a closed interval by repeatedly dividing the interval in half. Open methods like Newton-Raphson and secant methods use information about the nonlinear function to iteratively refine the estimated root without being restricted to an interval. The document also covers methods for equations with multiple roots like Muller's method.
2 1 polynomials
The document discusses identifying and evaluating polynomial functions. It defines polynomial functions as functions with whole number exponents and real number coefficients, and no variables in denominators. Polynomials can be written in standard form showing the degree and coefficients. The key points covered are: identifying the degree of a polynomial, finding the zeros of a polynomial function, directly substituting values for x, and using synthetic substitution to evaluate polynomials for any value of x.
BP106RMT.pdf
This document provides information about the key concepts and formulas related to differential calculus. It defines the derivative as the rate of change of a function with respect to an independent variable. Some common derivative formulas are presented for various standard functions like polynomial, exponential, logarithmic and trigonometric functions. Examples are given to demonstrate calculating derivatives using the formulas. The concepts of maxima and minima of functions are also introduced along with an example to find the maximum and minimum values of a given function without using derivatives.
The remainder theorem powerpoint
The document explains the Remainder Theorem in multiple ways using different examples and proofs. It states that the Remainder Theorem provides a test to determine if a polynomial f(x) is divisible by a polynomial of the form x-c. It proves that the remainder obtained when dividing f(x) by x-c is equal to the value of f(x) when x is substituted with c. It provides multiple examples working through applying the Remainder Theorem to determine if various polynomials are divisible.
3.3 Zeros of Polynomial Functions
This document discusses key concepts related to finding zeros of polynomial functions including:
1) The factor theorem, which states that a polynomial x - k is a factor of a function f(x) if and only if f(k) = 0.
2) The rational zeros theorem, which gives possible rational zeros based on the factors of the leading coefficient and constant term.
3) The fundamental theorem of algebra, which states that every polynomial of degree n has n complex zeros and examples of finding functions based on given zeros.
4) The conjugate zeros theorem, which states that if z = a + bi is a zero, then z = a - bi is also a zero for polynomials with real coefficients.
5
polynomials_.pdf
This PowerPoint presentation covers polynomials, including:
- Definitions of polynomials, monomials, binomials, trinomials, and the degree of a polynomial.
- The geometric meaning of zeros of polynomials - linear polynomials have one zero, quadratics have up to two zeros, and cubics have up to three.
- The relationship between the zeros and coefficients of a quadratic polynomial - the sum of the zeros equals the negative of the coefficient of x divided by the coefficient of x^2, and the product of the zeros equals the constant term divided by the coefficient of x^2.
- The division algorithm for polynomials - any polynomial p(x) can be divided by a non-zero polynomial
3.2 properties of division and roots t
The document discusses the Remainder Theorem and using it to find the value of a polynomial P(x) at a given point c. It provides examples of using synthetic division and the Factor Theorem to factor polynomials and determine if (x-c) is a factor based on whether c is a root. Exercises provide additional problems applying these concepts to find remainders, factor polynomials, and solve for constants.
Algebra
The document discusses several topics in algebra including:
1. Indices laws including am x an = am + n, am ÷ an = am - n, and (am)n = amn. Negative and fractional indices are also discussed.
2. Logarithms including the definition that logarithm of 'x' to base 'a' is the power to which 'a' must be raised to give 'x'. Change of base formula is also provided.
3. Series including the definition of finite and infinite series. Notation of sigma notation ∑ is introduced to represent the sum of terms.
Maths portfolio manvi
Detailed ppt on the topic polynomials, which will help in the revision for exam. Also for the ideas of making mathematics portfolio.
Logarithm
1. The document defines logarithmic functions and their relationship to exponential functions. Logarithmic functions are the inverse of exponential functions with the same base.
2. Methods for determining the logarithm of a number are presented, including using a logarithm table to find the logarithm of numbers that are not perfect powers of their base.
3. The laws of logarithms are proved using properties of exponents, including laws such as log(ab) = log(a) + log(b) and log(a/b) = log(a) - log(b). Examples of applying the laws of logarithms to equations are provided.
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Syllabus
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Chapter 2
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Chapter 3
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Chapter 4
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Chapter 5
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Chapter 6
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Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
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2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
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Factor theorem solving cubic equations
- 1. Polynomials and Partial Fractions In this lesson, you will learn that the factor theorem is a special case of the remainder theorem and use it to find factors of polynomials. 4.5 Factor Theorem Objectives
- 2. In the previous lesson, we saw that the Remainder Theorem can be used to find the remainder when a polynomial is divided by a linear divisor. We will use the factor theorem to find factors of polynomials. Polynomials and Partial Fractions The Factor Theorem is a special case of the Remainder Theorem when the remainder is zero: We also say that P( x ) is exactly divisible by x – a.
- 3. Since P(2) ≠ 0 Polynomials and Partial Fractions By the factor theorem, x + 1 is a factor of P( x ) Substitute for – 1 in P( x ). By the factor theorem, x – 2 is not a factor of P( x ). Substitute for 2 in P( x ). Example .
- 4. Equate coefficients to find the third factor. Polynomials and Partial Fractions Factorise the quadratic equation. Multiply the second equation by 16 and subtract. P(4) = 0; P( – 1) = 0. Substitute x = 4 and x = – 1. Example
- 5. Polynomials and Partial Fractions In this lesson, you will learn to apply the factor theorem to solving cubic equations of the form px 3 + qx 2 + rx + s = 0, where p , q , r and s are constants. 4.6 Solving Cubic Equations Objectives
- 6. We know how to solve linear equations and quadratic equations. We will use the factor theorem to help solve cubic equations. Polynomials and Partial Fractions Cubic equations can be solved by first applying the Factor Theorem to find one of the factors and then reducing the equation to a quadratic equation.
- 7. Polynomials and Partial Fractions By the factor theorem, x – 1 is a factor of P( x ) a = 3, – 1 × c = – 2 so c = 2 A cubic is a linear factor times a quadratic. choose α = 1 Since P(1) = 0 Equate coefficients of x 9 = – 1 × b + 2 b = – 7 Factorise the quadratic equation Example
- 8. a = 1, – 2 × c = 6 so c = – 3 Polynomials and Partial Fractions By the factor theorem, x – 2 is a factor of P( x ) A cubic is a linear factor times a quadratic – α × c = 6 so α is a factor of 6 choose α = 2 Since P(2) = 0 Equate coefficients of x 1 = – 2 × b – 3 b = – 2 Factorise the quadratic Example
Editor's Notes
- 4.5 Factor Theorem Objectives In this lesson you will learn that the factor theorem is a special case of the remainder theorem and use it to find factors of polynomials.
- Exercise 4.5, Page 90, Question 1
- Exercise 4.5, Page 90, Question 5
- 4.6 Solving Cubic Equations Objectives In this lesson you will learn to apply the factor theorem to solving cubic equations of the form px 3 + qx 2 + rx + s = 0, where p , q , r and s are constants.
- Exercise 4.6, Page 93, Question 1
- Exercise 4.6, Page 93, Question 3 (a)