Skip to main content
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 16
Chapter 4, Problem 16

Divide using long division. State the quotient, and the remainder, r(x).2x5−8x4+2x3+x22x3+1\(\frac{2x^5 - 8x^4 + 2x^3 + x^2}{2x^3 + 1}\)

Verified step by step guidance
1
Write the division in long division format, placing the dividend \(2x^{5} + 9x^{3} + 3x^{2}\) under the division bar and the divisor \(3x^{2} + 1\) outside.
Divide the leading term of the dividend, \$2x^{5}\(, by the leading term of the divisor, \)3x^{2}$, to find the first term of the quotient: \(\frac{2x^{5}}{3x^{2}} = \frac{2}{3}x^{3}\).
Multiply the entire divisor \(3x^{2} + 1\) by the term \(\frac{2}{3}x^{3}\) and subtract the result from the dividend to find the new remainder.
Repeat the process: divide the leading term of the new remainder by \$3x^{2}$ to find the next term of the quotient, multiply the divisor by this term, and subtract again.
Continue this process until the degree of the remainder is less than the degree of the divisor. The quotient is the sum of all terms found, and the remainder is the final expression left.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6m
Was this helpful?

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, and repeating until the degree of the remainder is less than the divisor.
Recommended video:
Guided course
05:13
Introduction to Polynomials

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 that of the divisor.
Recommended video:
Guided course
05:16
Standard Form of Polynomials

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 steps, and the remainder is the leftover polynomial with a degree less than the divisor, often expressed as r(x).
Recommended video:
Guided course
05:13
Introduction to Polynomials
Related Practice
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. 3x2+16x<−53x^2+16x<−5

526
views
Textbook Question

In Exercises 15–18, use the Leading Coefficient Test to determine the end behavior of the graph of the given polynomial function. Then use this end behavior to match the polynomial function with its graph. [The graphs are labeled (a) through (d).] <IMAGE>

f(x)=−x4+x2f(x)=−x^4+x^2

726
views
Textbook Question

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

1559
views
Textbook Question

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=(x−4)2−1

1309
views
Textbook Question

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

549
views
Textbook Question

Find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=−2(x−1)(x+2)2(x+5)3f(x)=-2(x-1)(x+2)^2(x+5)^3

826
views