2.4.5

Rewriting Rational Expressions

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Rewriting Rational Expressions

We can use everything we have learned about dividing polynomials to rewrite rational expressions.

Illustrative background for Rational ExpressionIllustrative background for Rational Expression ?? "content

Rational Expression

  • A rational expression is an expression of the form p(x)q(x)\frac{p(x)}{q(x)}, where p(x)p(x) and q(x)q(x) are polynomials and q(x)0q(x)\ne0.
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Division

  • We've already looked at the decomposition of a polynomial in terms of its quotient q(x)q(x), divisor (xc)(x-c), and remainder rr:
    • f(x)=q(x)(xc)+rf(x)=q(x)(x-c)+r
  • We can generalize the decomposition for any divisor d(x)d(x) that gives remainder r(x)r(x):
    • f(x)=q(x)d(x)+r(x)f(x)=q(x)d(x)+r(x)
  • Where r(x)r(x) is a polynomial of a degree less than that of d(x)d(x).
Illustrative background for Rewriting rational expressionsIllustrative background for Rewriting rational expressions ?? "content

Rewriting rational expressions

  • Dividing this equation by d(x)d(x) on both sides gives the following:
    • f(x)d(x)=q(x)+r(x)d(x)\frac{f(x)}{d(x)} = q(x) + \frac{r(x)}{d(x)}
  • Where d(x)0d(x)\ne0.
  • We can use this to rewrite rational expressions.
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Example

  • For example, x2+5x+6x+2\frac{x^2+5x+6}{x+2} can be rewritten as x+3x+3.
  • This is because x+2x+2 is a factor of the numerator, and so in this case r(x)=0r(x) = 0.
  • Here x2x\ne-2, as this would make the denominator equal to zero and the rational expression would be undefined.

Jump to other topics

1Proof

2Algebra & Functions

2.1Powers & Roots

2.2Quadratic Equations

2.3Inequalities

2.4Polynomials

2.5Graphs

2.6Functions

2.7Transformation of Graphs

2.8Partial Fractions (A2 Only)

3Coordinate Geometry

4Sequences & Series

5Trigonometry

6Exponentials & Logarithms

7Differentiation

8Integration

9Numerical Methods

10Vectors

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