Rewrite Rational Expressions (Calculator)

Related Pages
Polynomial Long Division
Synthetic Division
Common Core Algebra
Common Core Mathematics

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 out our new and fun Fraction Concoction Game.

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