Rewrite Rational Expressions (Calculator)

These lessons, with videos, examples and step-by-step solutions, help High School students learn how to rewrite simple rational expressions in different forms; write  a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x),  b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of  b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Suggested Learning Targets

• Rewrite rational expressions,
a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) by using inspection, long division, or synthetic division and comparing coefficients.

• Use a computer algebra system for complicated examples to assist with building a broader conceptual understanding.

Common Core: HSA-APR.D.6

Computer Algebra System

Polynomial Division using a computer algebra system (for example Wolfram Alpha)

TI-Nspire CX CAS Polynomial Division Expand function.

Comparing Coefficients

Example:
Divide x3 + x2 − 7 by x − 3 by comparing coefficients.

Rewrite the expression
x3 + x2 − 7 ≡ (Ax2 + Bx + C)(x − 3) + D

Expanding the right side we get,
(Ax2 + Bx + C)(x − 3) + D = Ax3 + (−3A + B)x2 + (−3B + C)x − 3C + D

Matching coefficients of the various powers of x on the left and right hand sides, we get
x3 + x2 − 7 ≡ Ax3 + (−3A + B)x2 + (−3B + C)x − 3C + D

A = 1
(−3A + B) = 1 ⇒ B = 4
(−3B + C) = 0 ⇒ C =12
− 3C + D = -7 ⇒ D = 29

The expression can then be written as
x3 + x2 − 7 ≡ (x2 + 4x + 12)(x − 3) + 29

Thus, (x3 + x2 − 7)/(x − 3) = x2 + 4x + 12 + 29/(x − 3)

Polynomial Division & Equating Coefficients

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.