12. Dividing Polynomials and Simplifying Rational Expressions

Photo Of Person Teaching On White Board by Vanessa Garcia is used under the pexels license.

Learning Objective:

Divide two polynomials to obtain a rational expression. Take account of the fact that a rational expression is undefined when (if) the polynomial in the denominator is zero. Apply factoring, if possible, to simplify a rational expression.

Video:
Interactive Activity

PDF Notes Link:

Dividing Polynomials Simplifying Rational ExpressionsDownload

Notes:

Dividing polynomials to obtain a rational expression

If we multiply the two polynomials, 2x² + 3 and x – 4, we obtain another polynomial, 2x³ – 8x² + 3x – 12.

However, when we divide two polynomials, we generally don’t obtain another polynomial. Rather, we obtain another type of algebraic expression called a rational expression.

For example, (2x² + 3) ÷ (x – 4) = (2x²+3) / (x-4). One thing to remember about a rational expression is that it is not defined when the denominator is 0, in this case when x = 4.

Simplifying rational expressions

We can’t simplify the rational expression (2x²+3) / (x-4). However, sometimes we can apply ideas about factoring polynomials to simplify a rational expression.

For example,

\frac{2x^3 + 6x^2}{2x^2} = \frac{\cancel{2x^2}(x+3)}{\cancel{2x^2}} = x+3.

However, we must remember that this expression is not defined when 2X2 = 0, i.e., when x = 0.

More Examples

\frac{x^2 - 4}{x^2 + 3x - 10} = \frac{(x+2)\cancel{(x-2)}}{(x+5)\cancel{(x-2)}} = \frac{x+2}{x+5}, \quad \{\forall x \ne -5, 2\}

\frac{x^2 + x - 12}{x^2 - 6x + 9} = \frac{(x+4)\cancel{(x-3)}}{(x-3)\cancel{(x-3)}} = \frac{x+4}{x-3}, \quad \{\forall x \ne 3\}

\frac{(x^2 + 4x + 4)(2x+2)}{(x^2 - 1)(x+2)} = \frac{(x+2)^{\cancel{2}} \ (2)\cancel{(x+1)}}{\cancel{(x+1)}(x-1)\cancel{(x+2)}} = \frac{2(x+2)}{x-1}, \quad \{\forall x \ne -2, -1, 1\}

\frac{3x^3 + 3x^2 - 18x}{6x^2 - 12x} = \frac{3x(x^2 + x - 6)}{6x(x-2)} = \frac{3\cancel{x}(x+3)\cancel{(x-2)}}{\cancel{6}^2 \cancel{x}\cancel{(x-2)}} = \frac{x+3}{2}, \quad {\forall x \ne 0, 2}

\frac{(2x-6)(3x^2 + 2x)}{(3x+2)(x^2 - 9)} = \frac{2\cancel{(x-3)}x\cancel{(3x+2)}}{\cancel{(3x+2)}(x+3)\cancel{(x-3)}} = \frac{2x}{x+3}, \quad \{\forall x \ne -\frac{2}{3}, -3, 3\}

Transcript:

The video transcripts are accessible for viewing and downloading below.

Dividing-Polynomials-and-Simplifying-Rational-ExpressionsDownload

Video Link:

https://media.tru.ca/media/Math+12/0_brngh213

Print Friendly, PDF & Email

Similar Posts