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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 35
Chapter 4, Problem 35

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

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Recall the Remainder Theorem: For a polynomial function ƒ(x), the remainder when ƒ(x) is divided by (x - k) is equal to ƒ(k). This means to find ƒ(k), you simply substitute k into the polynomial.
Write down the given polynomial function: \(ƒ(x) = 2x^2 - 3x - 3\) and the value \(k = 2\).
Substitute \(x = 2\) into the polynomial: \(ƒ(2) = 2(2)^2 - 3(2) - 3\).
Simplify the expression step-by-step: First calculate the square, then multiply, and finally perform the addition and subtraction.
The simplified result after substitution is the value of \(ƒ(2)\), which is the remainder when dividing by \((x - 2)\) according to the Remainder Theorem.

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Key Concepts

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

Polynomial Functions

A polynomial function is an expression consisting of variables and coefficients combined using addition, subtraction, and multiplication, with non-negative integer exponents. Understanding the structure of polynomials helps in evaluating them at specific values and applying related theorems.
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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 allows us to find the value of the polynomial at x = k by simply evaluating ƒ(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 variable with that value and simplifying. This process is essential for applying the Remainder Theorem and finding the remainder or function value efficiently.
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Guided course
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Introduction to Polynomials
Related Practice
Textbook Question

Solve each problem. Is x+1 a factor of ƒ(x)=x3+2x2+3x+2?

Textbook Question

Solve each problem. The speed of a pulley varies inversely as its diameter. One kind of pulley, with diameter 3 in., turns at 150 revolutions per minute. Find the speed of a similar pulley with diameter 5 in.

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Textbook Question

Solve each problem. Use Descartes' rule of signs to determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of ƒ(x)=x3+3x2−4x−2ƒ(x)=x^3+3x^2-4x-2.

Textbook Question

Graph each polynomial function. Factor first if the polynomial is not in factored form. ƒ(x)=(3x-1)(x+2)2

Textbook Question

Graph each quadratic function. Give the (a) vertex, (b) axis, (c) domain, and (d) range. ƒ(x) = -3x2 + 24x - 46

Textbook Question

Match the rational function in Column I with the appropriate description in Column II. Choices in Column II can be used only once. ƒ(x)=(x2+3x+4)/(x-5)