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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 19
Chapter 4, Problem 19

Divide using synthetic division. (3x2+7x−20)÷(x+5)

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1
Identify the divisor and rewrite it in the form \( x - c \). Since the divisor is \( x + 5 \), rewrite it as \( x - (-5) \), so \( c = -5 \).
Write down the coefficients of the dividend polynomial \( 3x^2 + 7x - 20 \). These are \( 3, 7, \) and \( -20 \).
Set up the synthetic division by placing \( c = -5 \) to the left and the coefficients \( 3, 7, -20 \) to the right.
Perform synthetic division steps: bring down the first coefficient, multiply it by \( c \), add to the next coefficient, and repeat until all coefficients are processed.
Interpret the final row of numbers as the coefficients of the quotient polynomial and the remainder. The quotient will have one degree less than the original 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 polynomials, making calculations faster and less error-prone.
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Polynomial Division

Polynomial division involves dividing one polynomial by another, similar to numerical division. Understanding how to divide polynomials helps in simplifying expressions, finding factors, and solving polynomial equations.
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Coefficients and Remainders

In synthetic division, only the coefficients of the dividend polynomial are used, and the remainder is the final value obtained after the division process. Recognizing how to interpret these coefficients and the remainder is essential for writing the quotient and remainder correctly.
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Related Practice
Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x)=x3−x2−9x+9f(x) = x^3 - x^2 - 9x + 9

Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 15–20.

As x→∞, f(x)→x\(\to\]\infty\),\(\text{ }\)f(x)\(\to\)_____

Textbook Question

Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial function. f(x)=5x3+7x2−x+9f(x)=5x^3+7x^2−x+9

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Textbook Question

In Exercises 19–24, (a) Use the Leading Coefficient Test to determine the graph's end behavior. (b) Determine whether the graph has y-axis symmetry, origin symmetry, or neither. (c) Graph the function. f(x)=4x−x3f(x) = 4x - x^3

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Textbook Question

In Exercises 17–24, a) List all possible rational roots. b) List all possible rational roots. c) Use the quotient from part (b) to find the remaining roots and solve the equation. x3−10x−12=0

Textbook Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x2−4x≥0x^2−4x≥0