Textbook Question
Solve each quadratic inequality. Give the solution set in interval notation. x2 - 2 > x
Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
Chapter 4, Problem 20
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.
Verified step by step guidance1
Identify the divisor polynomial and rewrite it in the form \(x - c\). Since the divisor is \(x + 4\), rewrite it as \(x - (-4)\), so \(c = -4\).
Apply the Factor Theorem by evaluating the first polynomial at \(x = -4\). This means substituting \(-4\) into \(5x^4 + 16x^3 - 15x^2 + 8x + 16\) and calculating the result.
If the result from step 2 is zero, then \(x + 4\) is a factor of the polynomial. If not, it is not a factor.
To confirm, perform synthetic division of the polynomial \(5x^4 + 16x^3 - 15x^2 + 8x + 16\) by \(x + 4\) using \(c = -4\). Set up the synthetic division with coefficients [5, 16, -15, 8, 16].
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 tell you if \(x + 4\) is a factor (remainder zero) or not (non-zero remainder).
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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 + 4 is a factor, substitute -4 into the polynomial and see if the result is zero. If it is, then x + 4 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. The remainder helps verify if 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
Write each formula as an English phrase using the word varies or proportional. C=2πr, where C is the circumference of a circle of radius r.
Textbook Question
Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first.
Textbook Question
If ƒ(x) is a polynomial function with real coefficients, and if 7+2i is a zero of the function, then what other complex number must also be a zero?
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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