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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 57
Chapter 4, Problem 57

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 - 2x + 2; k = 1-i

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1
Write down the coefficients of the polynomial ƒ(x) = x^2 - 2x + 2. These are 1 (for x^2), -2 (for x), and 2 (constant term).
Set up synthetic division using the coefficients [1, -2, 2] and the divisor k = 1 - i. Remember, synthetic division works with numbers, so treat k as a complex number.
Bring down the first coefficient (1) as it is. Then multiply this number by k = 1 - i, and write the result under the next coefficient.
Add the second coefficient (-2) and the product from the previous step. Write the sum below the line. Repeat the multiply and add process for the last coefficient.
The final number you get after the last addition is the remainder, which equals ƒ(k). If this remainder is 0, then k is a zero of the polynomial. If not, this remainder is the value of ƒ(k).

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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 division process by using only the coefficients of the polynomial, making it faster and less error-prone than long division. It also helps evaluate the polynomial at k to check if k is a root.
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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 = 1 - i.
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Dividing Complex Numbers

Zeros of a Polynomial

A zero (or root) of a polynomial is a value k for which the polynomial evaluates to zero, i.e., f(k) = 0. Determining whether a given number is a zero involves substituting it 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

For each polynomial function, identify its graph from choices A–F. ƒ(x)=-(x-2)(x-5)

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Textbook Question

Find a polynomial function ƒ(x) of degree 3 with real coefficients that satisfies the given conditions. Zero of -3 having multiplicity 3; ƒ(3)=36

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Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=2x5-x4+2x3-2x2+4x-4; no real zero greater than 1

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Textbook Question

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

Textbook Question

Solve each rational inequality. Give the solution set in interval notation. 8 /(x - 2) ≥ 2

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Textbook Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x4-x3+3x2-8x+8; no real zero greater than 2