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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 59
Chapter 4, Problem 59

Use synthetic division to determine whether the given number k is a zero of the polynomial function. If it is not, give the value of ƒ(k). ƒ(x) = x2 + 3x + 4; k = 2+i

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Identify the polynomial function and the value of \( k \). Here, \( f(x) = x^2 + 3x + 4 \) and \( k = 2 + i \), where \( i \) is the imaginary unit.
Set up synthetic division using \( k = 2 + i \) as the divisor. Write down the coefficients of \( f(x) \): 1 (for \( x^2 \)), 3 (for \( x \)), and 4 (constant term).
Begin synthetic division by bringing down the first coefficient (1) as is. Then multiply this by \( k = 2 + i \) and add the result to the next coefficient (3).
Continue the process: multiply the new sum by \( k = 2 + i \) and add it to the last coefficient (4). The final number you get is the remainder, which equals \( f(k) \).
If the remainder is zero, then \( k \) is a zero of the polynomial. If not, the remainder is the value of \( f(k) \), showing that \( k \) is not a zero.

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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 factor of the form (x - k). It simplifies the long division process by using only the coefficients of the polynomial and the value k, making it faster to evaluate polynomials or find remainders.
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Higher Powers of i

Complex Numbers

Complex numbers include a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with i² = -1. Understanding how to perform arithmetic with complex numbers is essential when evaluating polynomials at complex values like k = 2 + i.
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Dividing Complex Numbers

Zeros of a Polynomial

A zero of a polynomial is a value k for which the polynomial evaluates to zero, meaning f(k) = 0. Determining whether k is a zero involves substituting k into the polynomial or using synthetic division to check if the remainder is zero.
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Finding Zeros & Their Multiplicity
Related Practice
Textbook Question

Solve each rational inequality. Give the solution set in interval notation. (2x - 3)/(x + 1) > 4

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=3x4+2x3-4x2+x-1; no real zero greater than 1

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Identify any vertical, horizontal, or oblique asymptotes in the graph of y=ƒ(x). State the domain of ƒ.

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Solve each rational inequality. Give the solution set in interval notation. (x - 8)/(x - 4) < 3

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. See Example 6.

ƒ(x)=x5+2x3−2x2+5x+5ƒ(x)=x^5+2x^3-2x^2+5x+5; no real zero less than -1

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x4+x3-x2+3; no real zero less than -2

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