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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblem 39
Chapter 4, Problem 39

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x4−5x3−x2+3x+2; f(−1/2)

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Identify the polynomial function and the value at which you need to evaluate it: \(f(x) = 2x^{4} - 5x^{3} - x^{2} + 3x + 2\) and you want to find \(f\left(-\frac{1}{2}\right)\).
Set up synthetic division using the divisor \(x - r\), where \(r = -\frac{1}{2}\). Write the coefficients of the polynomial in descending order of powers: \(2, -5, -1, 3, 2\).
Perform synthetic division by bringing down the first coefficient, then multiply it by \(r = -\frac{1}{2}\), add this result to the next coefficient, and repeat this process for all coefficients.
The final number you get after completing synthetic division is the remainder, which by the Remainder Theorem equals \(f\left(-\frac{1}{2}\right)\).
Interpret this remainder as the value of the function at \(x = -\frac{1}{2}\), completing the evaluation without directly substituting into the polynomial.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 of the polynomial, making calculations faster and less error-prone. This method is especially useful for evaluating polynomials and finding remainders.
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Remainder Theorem

The Remainder Theorem states that when a polynomial f(x) is divided by (x - c), the remainder is equal to f(c). This means you can find the value of the polynomial at x = c by performing synthetic division and looking at the remainder, providing a quick way to evaluate polynomials without direct substitution.
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Polynomial Evaluation

Polynomial evaluation involves finding the value of a polynomial function at a specific input. Instead of substituting the value directly into the polynomial expression, synthetic division combined with the Remainder Theorem offers an efficient alternative, especially for higher-degree polynomials, to determine f(c) quickly.
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Related Practice
Textbook Question

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. f(x)=x3−4x2−7x+10

Textbook Question

In Exercises 33–40, use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=x5−x3−1; between 1 and 2

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Textbook Question

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=x4+5x3+5x2−5x−6;f(3)

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Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x3+x2+4x+4>0x^3+x^2+4x+4>0

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Textbook Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. g(x)=12x2/(3x2+1)

Textbook Question

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