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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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.
Finding values of polynomial functions
This document discusses using synthetic division and the remainder theorem to find the value of polynomial functions at given points. It provides examples of using both synthetic division and the remainder theorem to find the value of polynomials like P(x) = 2x^3 - 8x^2 + 19x - 12 at x = 3. The key points are that the remainder R obtained from synthetic division gives the value of the polynomial function at the given point, f(c), and that if R = 0, then x - c is a factor of the polynomial. Exercises are provided to have students practice finding polynomial values using these methods.
Chapter 5 Polynomials
1) The document discusses polynomials, including adding, subtracting, multiplying, and dividing polynomials. It also covers finding the roots and zeros of polynomials.
2) Partial fraction decomposition is explained for denominators that are linear factors, repeated linear factors, quadratic factors, and repeated quadratic factors.
3) Key concepts covered include the remainder and factor theorems, zeros of polynomials, and converting improper rational expressions to proper rational expressions using long division and partial fraction decomposition.
Appendex b
This document provides an overview of solving polynomial equations. It defines polynomials and their key properties like degree, coefficients, and roots. It introduces several theorems for finding roots, including the Remainder Theorem, Factor Theorem, and the idea that a polynomial of degree n has n roots when counting multiplicities. Methods discussed include factoring, long division, and the quadratic formula. The document explains it is not possible to express solutions of polynomials of degree 5 or higher using radicals.
Polynomials by nikund
This ppt will help you knowing more about different types of polynomials and will help you to understand better.
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The document describes aspects of polynomials including:
- Definitions of polynomials and their components
- Algebraic operations on polynomials like addition, subtraction, and multiplication
- Dividing polynomials using long division and synthetic (Horner's) division methods
- The remainder theorem stating the remainder of dividing a polynomial by (x - k) is the value of the polynomial at k
- The factorization theorem relating the factors of a polynomial to its roots
- Techniques for factorizing polynomials based on the sums of coefficients
The document contains examples and explanations of key polynomial concepts over multiple pages.
Polynomials
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.
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.
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.
Module in Remainder Theorem
The document discusses the Remainder Theorem, which states that the remainder obtained when dividing a polynomial f(x) by a linear polynomial x-c is equal to the value of f(x) when x is substituted with c. It provides examples to show how to use the Remainder Theorem to determine whether a given polynomial is divisible by x-c. The document also contains practice exercises for readers to apply the Remainder Theorem.
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.
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.
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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)