Chapter 11.2, Problem 10LC

(a)

To determine

Explain how to multiply a rational expression by a polynomial.

Answer to Problem 10LC

Explanation of Solution

Given:

Explain how to multiply a rational expression by a polynomial.

Concept Used:

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

An algebraic expression where both the numerator and the denominator are polynomials e.g $\frac{x-4}{x}$ is called a rational expression.

Since the denominator can't be zero there are values of x which are excluded from the rational expression. The expression above has an excluded value of zero.

Calculation:

To multiply polynomial, first find the factor both the numerator and denominator of both the expressions and then multiply the remaining polynomial.

Steps to multiply the polynomial fractions

Factor each the numerators and denominators of all fractions completely.

Cancel or reduce the fractions.

Rewrite the remaining factor.

Example: $\frac{y^2-9}{y^2-3y}·\frac{y}{y^2+9y+18}$

Step 1: Factor both the numerator and denominator.

  $\frac{y^2-9}{y^2-3y}·\frac{y}{y^2+9y+18}=\frac{(y+3)(y-3)}{y(y-3)}·\frac{y}{(y+3)(y+6)}$

Now cancel out the common factors from numerator and denominator.

  $\begin{array}{l}\frac{y^2-9}{y^2-3y}·\frac{y}{y^2+9y+18}\\ \frac{(y+3)(y-3)}{y(y-3)}·\frac{y}{(y+3)(y+6)}=\frac{y+3}{y+3}·\frac{y-3}{y-3}·\frac{y}{y}·\frac{1}{y+6}=\frac{1}{y+6}\end{array}$

Simplified the multiplication of two rational expression is $\frac{y^2-9}{y^2-3y}·\frac{y}{y^2+9y+18}=\frac{1}{y+6}$

(b)

To determine

Explain how to divide a rational expression by a polynomial.

Answer to Problem 10LC

Explanation of Solution

Given:

Explain how to divide a rational expression by a polynomial.

Concept Used:

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

An algebraic expression where both the numerator and the denominator are polynomials e.g $\frac{x-4}{x}$ is called a rational expression.

Since the denominator can't be zero there are values of x which are excluded from the rational expression. The expression above has an excluded value of zero.

Calculation:

Dividing Rational Expressions The method of dividing rational expressions is same as the method of dividing fractions. That is, to divide a rational expression by another rational expression, multiply the first rational expression by the reciprocal of the second rational expression.

For all rational expressions: $\frac{a}{b}\text{and}\frac{c}{d}$ with $b\ne 0;c\ne 0\text{and}d\ne 0$ .and $\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}=\frac{ad}{bc}$

Example:

Divide and then simplify:

  $\frac{x^2-4}{x+6}÷\frac{x+2}{2x+12}$

Write as multiplication by the reciprocal of the 2^nd fraction.

The reciprocal of $\frac{x+2}{2x+12}$ is $\frac{2x+12}{x+2}$

  $\frac{x^2-4}{x+6}÷\frac{x+2}{2x+12}=\frac{x^2-4}{x+6}·\frac{2x+12}{x+2}$

Find the factors of numerator and denominator:

  $\frac{x^2-4}{x+6}·\frac{2x+12}{x+2}=\frac{(x+2)(x-2)}{x+6}·\frac{2(x+6)}{x+2}$

Cancel out the common factors:

  $\frac{(x+2)(x-2)}{x+6}·\frac{2(x+6)}{x+2}=\frac{x+2}{x+2}·\frac{x+6}{x+6}·\frac{2(x-2)}{1}$

Simplify

  $\frac{2(x-2)}{1}=2x-4$

  $\frac{x^2-4}{x+6}÷\frac{x+2}{2x+12}=2x-4$