 # Rewrite Rational Expressions (Calculator)

Videos and lessons with examples and solutions 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

## 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+x27 by  x3 by comparing coefficients.

Rewrite the expression
x3+x27(Ax2+Bx+C )(x3)+D

Expanding the right side we get,
(Ax2+Bx+C )(x3)+D = Ax3 + (3A+B)x2+(3B+C )x3C + D

Matching coefficients of the various powers of x on the left and right hand sides, we get
x
3
+x27 ≡ Ax3 + (3A+B)x2+(3B+C )x3C + 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+x27(x2+4x+12)(x3)+29

Thus, (x3+x27)/( x3) = (x2+4x+12) + 29/( x3)

Polynomial Division & Equating Coefficients

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