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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 40
Chapter 4, Problem 40

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=6x4+10x3+5x2+x+1; f(−2/3)

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Identify the polynomial function and the value at which you need to evaluate it: \(f(x) = 6x^{4} + 10x^{3} + 5x^{2} + x + 1\) and you want to find \(f\left(-\frac{2}{3}\right)\).
Set up synthetic division using the value \(-\frac{2}{3}\) as the divisor. Write down the coefficients of the polynomial in descending order of powers: 6, 10, 5, 1, and 1.
Perform synthetic division by bringing down the first coefficient (6), then multiply it by \(-\frac{2}{3}\), add the result to the next coefficient, and continue this process across all coefficients.
The final number you obtain after completing synthetic division is the remainder, which by the Remainder Theorem equals \(f\left(-\frac{2}{3}\right)\).
Interpret the remainder as the value of the function at \(x = -\frac{2}{3}\), 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 at specific values.
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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, which provides 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 given 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 compute 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

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Textbook Question

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. f(x)=3x3−8x2+x+2; between 2 and 3

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Textbook Question

For Exercises 40–46, (a) List all possible rational roots or rational zeros. (b) Use Descartes's Rule of Signs to determine the possible number of positive and negative real roots or real zeros. (c) Use synthetic division to test the possible rational roots or zeros and find an actual root or zero. (d) Use the quotient from part (c) to find all the remaining roots or zeros. f(x)=x3+3x2−4f(x) = x^3 + 3x^2 - 4

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Textbook Question

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=−4x2+8x−3

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Textbook Question

Find the horizontal asymptote, if there is one, of the graph of each rational function. h(x)=12x3/(3x2+1)

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Textbook Question

An equation of a quadratic function is given. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range. f(x)=3x2−12x−1

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