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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 9
Chapter 4, Problem 9

Divide using long division. State the quotient, and the remainder, r(x)r(x). 2x3+7x2+9x−20x+3\(\frac{2x^3 + 7x^2 + 9x - 20}{x + 3}\)

Verified step by step guidance
1
Identify the dividend and divisor. Here, the dividend is \(2x^{3} + 7x^{2} + 9x - 20\) and the divisor is \(x + 3\).
Set up the long division by writing \(2x^{3} + 7x^{2} + 9x - 20\) under the division bar and \(x + 3\) outside.
Divide the leading term of the dividend, \$2x^{3}\(, by the leading term of the divisor, \)x\(, to get the first term of the quotient: \)2x^{2}$.
Multiply the entire divisor \(x + 3\) by \$2x^{2}$ and subtract the result from the dividend to find the new polynomial to bring down.
Repeat the process: divide the new leading term by \(x\), multiply the divisor by this term, subtract, and continue until the degree of the remainder is less than the degree of the divisor.

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Key Concepts

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

Polynomial Long Division

Polynomial long division is a method used to divide one polynomial by another, similar to numerical long division. It involves dividing the leading term of the dividend by the leading term of the divisor, multiplying the divisor by this result, subtracting from the dividend, and repeating until the degree of the remainder is less than the divisor.
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Quotient and Remainder in Polynomial Division

When dividing polynomials, the quotient is the polynomial result of the division, and the remainder is what is left over with a degree less than the divisor. The division can be expressed as Dividend = Divisor × Quotient + Remainder, which helps in understanding the relationship between these components.
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Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. In polynomial division, the process continues until the degree of the remainder is less than the degree of the divisor, ensuring the division is complete and the remainder cannot be further divided.
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Related Practice
Textbook Question

Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.


As x→−3−x\(\to\)-3^{-}, f(x)→f\(\left\)(x\(\right\))\(\to\)____

Textbook Question

Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product?

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Textbook Question

In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x3+x2−4x−4

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Textbook Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2(x−3)2+1

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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. x2−6x+9<0

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Textbook Question

In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x1/3 −4x2+7