# Rewrite Rational Expressions (Calculator)

Videos and lessons to 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

Related Topics:

Polynomial Long Division

Synthetic Division

Common Core (Algebra)

Common Core for Mathematics

## 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

x^{3}+x^{2}−7 by

x−3 by comparing coefficients.

Rewrite the expression

x^{3}+x^{2}−7 ≡

(Ax^{2}+Bx+C )(x−3)+D
Expanding the right side we get,

(Ax^{2}+Bx+C )(x−3)+D = Ax

^{3} +

(−3A+B)x^{2}+(−3B+C )x−3C + D
Matching coefficients of the various powers of x on the left and right hand sides, we get

x^{3}+x^{2}−7 ≡ Ax

^{3} +

(−3A+B)x^{2}+(−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

x^{3}+x^{2}−7 ≡

(x^{2}+4

x+12

)(x−3)+29

Thus,

(x^{3}+x^{2}−7)/( x−3) =

(x^{2}+4

x+12

) + 29/

( x−3)

Polynomial Division & Equating Coefficients

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