Textbook Question
Graph each function. Determine the largest open intervals of the domain over which each function is (a) increasing or (b) decreasing. ƒ(x)=(1/2)(x-2)2+4
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 19
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first.
Verified step by step guidance1
Identify the divisor polynomial and rewrite it in the form \(x - c\). Since the divisor is \(x + 3\), rewrite it as \(x - (-3)\), so \(c = -3\).
Apply the Factor Theorem by evaluating the first polynomial at \(x = -3\). This means substituting \(-3\) into \(2x^4 + 5x^3 - 2x^2 + 5x + 6\) and calculating the result.
If the result from step 2 is zero, then \(x + 3\) is a factor of the polynomial. If not, it is not a factor.
To confirm, perform synthetic division of the first polynomial by \(x + 3\) using \(c = -3\). Set up the synthetic division with the coefficients of the polynomial: 2, 5, -2, 5, 6.
Carry out the synthetic division step-by-step: bring down the first coefficient, multiply by \(c\), add to the next coefficient, and repeat until all coefficients are processed. The remainder will be the last value obtained. If the remainder is zero, \(x + 3\) is a factor; otherwise, it is not.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factor Theorem
The Factor Theorem states that a polynomial f(x) has a factor (x - c) if and only if f(c) = 0. To check if a binomial like x + 3 is a factor, substitute -3 into the polynomial and see if the result is zero. If it is, then x + 3 divides the polynomial exactly.
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Synthetic Division
Synthetic division is a shortcut method for dividing a polynomial by a linear binomial of the form x - c. It simplifies the long division process by using only the coefficients, making it faster to find the quotient and remainder. If the remainder is zero, the divisor is a factor.
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Polynomial Factorization
Polynomial factorization involves expressing a polynomial as a product of its factors. Identifying factors helps simplify expressions and solve polynomial equations. Using the Factor Theorem and synthetic division together aids in breaking down complex polynomials into simpler components.
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Related Practice
Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation. x2 - 2 > x
Textbook Question
Use synthetic division to perform each division. (x4 - 3x3 - 4x2 + 12x) / x-2
Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation. 2x2 + 5 ≤ 11x
Textbook Question
Graph the following on the same coordinate system.
(a) y = x2
(b) y = 3x2
(c) y = 1/3x2
(d) How does the coefficient of x2 affect the shape of the graph?
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Textbook Question
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.