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 13

Chapter 4, Problem 13

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x³+13x²−11x−15)/(3x²−x−3)

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Identify the dividend and divisor: The dividend is $6x^{3} + 13x^{2} - 11x - 15$ and the divisor is $3x^{2} - x - 3$.

Set up the long division by writing the dividend under the division bar and the divisor outside.

Divide the leading term of the dividend, \$6x^{3}$, by the leading term of the divisor, $3x^{2}$, to find the first term of the quotient: $\frac{6x^{3}}{3x^{2}} = 2x$.

Multiply the entire divisor $3x^{2} - x - 3$ by \$2x$ and subtract the result from the dividend to find the new polynomial to bring down.

Repeat the process: divide the new leading term by \$3x^{2}$, 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.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the expression. Understanding the degree is essential in polynomial division because the division process continues until the remainder's degree is less than the divisor's degree, indicating the division is complete.

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 process, and the remainder is the leftover polynomial with a degree less than the divisor. Expressing the division as dividend = divisor × quotient + remainder is fundamental.

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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. 6x²+x>1

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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+x²−3x+1

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Textbook Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2x²−8x+3

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Textbook Question

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

Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as the cube root 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. 2x²+x<15

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In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x3+13x2−11x−15)/(3x2−x−3)

Verified video answer for a similar problem:

Key Concepts

Polynomial Long Division

Degree of a Polynomial

Quotient and Remainder in Polynomial Division

In Exercises 1–16, divide using long division. State the quotient, and the remainder, r(x). (6x³+13x²−11x−15)/(3x²−x−3)