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Ch. 3 - Polynomial and Rational Functions
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 3 - Polynomial and Rational FunctionsProblem 43
Chapter 4, Problem 43

For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = 2x5 - 10x3 - 19x2 - 50; k=3

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1
Recall the Remainder Theorem, which states that the remainder when a polynomial ƒ(x) is divided by (x - k) is equal to ƒ(k). So, to find ƒ(3), we simply evaluate the polynomial at x = 3.
Write down the polynomial function: \(ƒ(x) = 2x^{5} - 10x^{3} - 19x^{2} - 50\).
Substitute \(x = 3\) into the polynomial: \(ƒ(3) = 2(3)^{5} - 10(3)^{3} - 19(3)^{2} - 50\).
Calculate each term separately: compute \$2(3)^{5}\(, \)-10(3)^{3}\(, \)-19(3)^{2}\(, and \)-50$.
Add all the calculated terms together to find the value of \(ƒ(3)\), which is the remainder when dividing by \((x - 3)\).

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Key Concepts

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

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to whole-number exponents and coefficients combined using addition, subtraction, and multiplication. Understanding the structure of polynomials helps in evaluating them at specific values and applying theorems related to their behavior.
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Remainder Theorem

The Remainder Theorem states that when a polynomial ƒ(x) is divided by (x - k), the remainder is equal to ƒ(k). This theorem allows for quick evaluation of the polynomial at x = k without performing full polynomial division.
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Evaluating Polynomials at a Given Value

Evaluating a polynomial at a specific value involves substituting the value into the polynomial and simplifying. This process is essential for applying the Remainder Theorem and finding the remainder or value of the function at that point.
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Related Practice
Textbook Question

Solve each polynomial inequality. Give the solution set in interval notation. x4 + 2x3 + 36 < 11x2 + 12x

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For each polynomial function, use the remainder theorem to find ƒ(k). ƒ(x) = x2 + 4; k = 2i

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Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of each rational function. See Example 4.

ƒ(x)=(x2+4)/(x−1)ƒ(x)=(x^2+4)/(x-1)

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Solve each problem. Give the maximum number of turning points of the graph of each function. ƒ(x)=4x^3-6x^2+2

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Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=2x3-5x2-x+6

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Textbook Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. ƒ(x) = -2x2 - 8x - 7

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