Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 12

Chapter 4, Problem 12

Divide using long division. State the quotient, and the remainder, r(x). (x⁴−81)/(x−3)

Verified step by step guidance

Identify the dividend and divisor. Here, the dividend is $x^{4} - 81$ and the divisor is $x - 3$.

Set up the long division by writing $x^{4} - 81$ under the division bar and $x - 3$ outside the bar.

Divide the leading term of the dividend, $x^{4}$, by the leading term of the divisor, $x$, to get the first term of the quotient: $x^{3}$.

Multiply the entire divisor $x - 3$ by $x^{3}$ and subtract the result from the dividend to find the new polynomial to divide.

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, subtracting, and repeating until the degree of the remainder is less than the divisor.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree helps determine when to stop the division process, as the remainder must have a degree less than the divisor's degree.

Quotient and Remainder in Polynomial Division

When dividing polynomials, the result consists of a quotient and a remainder. The quotient is the polynomial obtained from the division, and the remainder is the leftover polynomial with a degree less than the divisor, often expressed as r(x).

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Identify which graphs are not those of polynomial functions.

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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)=2x³−3x²−11x+6

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Textbook Question

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

As $x \to - \infty, f (x) \to_{_{}}$ _____

Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube of z and inversely as y.

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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. 9x²+3x−2≥0

Divide using long division. State the quotient, and the remainder, r(x). (x4−81)/(x−3)

Verified video answer for a similar problem:

Key Concepts

Polynomial Long Division

Degree of a Polynomial

Quotient and Remainder in Polynomial Division

Divide using long division. State the quotient, and the remainder, r(x). (x⁴−81)/(x−3)