Multiplication of rational expressions

Let p (x) and q (x) be two polynomials, with q (x) ≠ 0, then the quotient p (x) / q (x) will be called  rational expression or algebraic expression. Now let p (x) / q (x) and r (x) / s (x) be two rational expressions with q (x) ≠ 0 and s (x) ≠ 0, then:

[p (x) / q (x)] × [r (x) / s (x)] = p (x) ∙ s (x) / q (x) ∙ (r (x)

In general, to multiply rational expressions, we recommend following these steps:

Steps to multiply rational expressions

  1. Having two terms in the form of a fraction, numerator by numerator and denominator by denominator are multiplied.
  2. The numerator and denominator are factored.
  3. The expression is simplified.

multiplication of rational expressions - formula

Example 1: Solve the following multiplication of rational expression: [(x + 3) / 2x] × [5x² / (x + 2)].

We multiply the numerators of both expressions, we do the same with the denominators:

[(x + 3) / 2x] × [5x² / (x + 2)] = [5x² (x + 3)] / [(2x) (x + 2)] =

We simplify:

= [5x (x + 3)] / [2 (x + 2)]

Final score:

[(x + 3) / 2x] × [5x² / (x + 2)] = [5x (x + 3)] / [2 (x + 2)]

Example 2: Solve the following multiplication of rational expression:

 [(y² - 4) / (y² + 5x + 4)] × [(y² + 2y - 8) / (y² - 4y + 4)].

[(y² - 4) / (y² + 5x + 4)] × [(y² + 2y - 8) / (y² - 4y + 4)] = [(y² - 4) [(y² + 2y - 8) / ( y² + 5x + 4) (y² - 4y + 4)] =

We factor:

= [(y + 2) (y - 2) (y + 4) (y - 2)] / [(y + 1) (y + 4) (y - 2) (y - 2)] =

We simplify the previous expression:

= (y + 2) / (y + 1)

Final score:

[(y² - 4) / (y² + 5x + 4)] × [(y² + 2y - 8) / (y² - 4y + 4)] = (y + 2) / (y + 1)