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# Multiplying Rational Expressions â€“ Techniques & Examples

To **learn how to multiply rational expressions**, letâ€™s first recall the **multiplication of numerical fractions.**

Multiplication of fractions involves separately finding the product of numerators and the product of denominators of given fractions.

For instance, if a/b and c/d are any two fractions, then;

a/b Ã— c/d = a Ã— c/b Ã— d. Letâ€™s take a look at the examples below:

- Multiply 2/7 by 3/5

__Solution__

2/7 Ã— 3/5

= 2 Ã— 3/7 Ã— 5= 6/35

- Multiply 5/9 by (-3/4)

__Solution__

5/9 Ã— (-3/4)

= 5 Ã— -3/9 Ã— 4

= -15/36

= -5/12

Similarly, rational expressions are multiplied by following the same rule.

## How to Multiply Rational Expressions?

**To multiply rational expressions, we apply the steps below:**

- Completely factor out denominators and numerators of both fractions.
- Cancel out common terms in the numerator and denominator.
- Now rewrite the remaining terms both in the numerator and denominator.

Use the algebraic identities below to help you in factoring the polynomials:

- (aÂ² – bÂ²) = (a + b) (a –Â b)
- (xÂ² – 4Â²) = (x + 4) (x – 4)
- (xÂ² – 2Â²) = (x + 2) (x – 2)
- (aÂ³ + bÂ³) = (a + b) (aÂ² – a b + bÂ²)

*Example 1*

SimplifyÂ (xÂ² – 2x) / (x + 2) *Â (3 x + 6)/ (x – 2)

__Solution__

Factor the numerators,

(xÂ² – 2x) / (x + 2) *Â (3 x + 6)/ (x – 2)

âŸ¹ x (x – 2) / (x + 2) *Â 3(x + 2)/ (x – 2)

Cancel out common terms in numerators and denominators of both fractions to get;

âŸ¹ 3x

*Example 2*

Solve [(x^{2} â€“ 3x â€“ 4)/ (x^{2} â€“ x -2)] * [(x^{2} â€“ 4)/ (x^{2} -+ x -20)]

__Solution__

First, factor the numerators and denominators of both fractions.

[(x â€“ 4) (x + 1)/ (x + 1) (x â€“ 2)] * [(x + 2) (x â€“ 2)/ (x â€“ 4) (x + 5)]

Cancel out common terms and rewrite the remaining terms

= x + 2/x + 5

*Example 3*

Multiply [(12x â€“ 4x^{2})/ (x^{2} + x â€“ 12)] * [(x^{2 }+ 2x – 8)/x^{3} â€“ 4x)]

__Solution__

Factor the rational expressions.

âŸ¹ [-4x (x â€“ 3)/ (x â€“ 3) (x + 4)] * [(x â€“ 2) (x + 4)/x (x + 2) (x â€“ 2)]

Reduce the fractions by cancelling common terms in the numerators and denominators to get;

= -4/x + 2

*Example 4*

Multiply [(2x^{2 }+ x â€“ 6)/ (3x^{2} â€“ 8x â€“ 3)] * [(x^{2} â€“ 7x + 12)/ (2x^{2} â€“ 7x â€“ 4)]

__Solution__

Factor the fractions

âŸ¹ [(2x â€“ 3) (x + 2)/ (3x + 1) (x â€“ 3)] * [(x â€“ 30(x â€“ 4)/ (2x + 1) (x â€“ 4)]

Cancel out common terms in the numerators and denominators and rewrite the remaining terms.

âŸ¹ [(2x â€“ 3) (x + 2)/ (3x + 1) (2x + 1)]

*Example 5*

Simplify [(xÂ² – 81)/ (xÂ² – 4)]Â *Â [(xÂ² + 6 x + 8)/ (xÂ² – 5 x – 36)]

__Solution__

Factor the numerators and denominators of each fraction.

âŸ¹ [(x + 9) (x â€“ 9)/ (x + 2) (x – 2)] * [(x + 2) (x + 4)/ (x – 9) (x + 4)]

On cancelling common terms, we get;

= (x + 9)/ (x – 2).

*Example 6*

Simplify [(xÂ² – 3 x – 10)/ (xÂ² – x – 20)]Â *Â [(xÂ² – 2 x + 4)/ (xÂ³ + 8)]

__Solution__

Factor out (xÂ³ + 8) using the algebraic identity (aÂ³ + bÂ³) = (a + b) (aÂ² – a b + bÂ²).

âŸ¹ (xÂ³ + 8) = (x + 2) (xÂ² – 2 x + 4).

âŸ¹ (xÂ² – 3 x – 10) = (x – 5) (x + 2)

âŸ¹ (xÂ² – x – 20) = (x – 5) (x + 4)

[(xÂ² – 3 x – 10)/ (xÂ² – x – 20)]Â *Â [(xÂ² – 2 x + 4)/ (xÂ³ + 8)] = [(x – 5) (x + 2)/ (x – 5) (x + 4)] * [(xÂ² – 2 x + 4)/ (x + 2) (xÂ² – 2 x + 4)]

Now, cancel out common terms to get;

= 1/ (x + 4).

*Example 7*

Simplify [(x + 7)/ (xÂ² + 14 x + 49)]Â *Â [(xÂ²Â + 8x + 7)/ (x + 1)]

__Solution__

Factor the fractions.

âŸ¹ (xÂ² + 14 x + 49) = (x + 7) (x + 7)

âŸ¹ (xÂ²Â + 8x + 7) = (x + 1) (x + 7)

= [(x + 7)/ (x + 7) (x + 7)]Â * [(x + 1) (x + 7)/ (x + 1)]

On cancelling common terms, we get the answer as;

= 1

*Example 8*

Multiply [(xÂ² – 16)/ (x – 2)] * [(xÂ² – 4)/ (xÂ³ + 64)]

__Solution__

Use the algebraic identity (aÂ² – bÂ²) = (a + b) (a – b) to factor (xÂ² – 16) and (xÂ² – 4).

(xÂ² – 4Â²) âŸ¹ (x + 4) (x â€“ 4)

(xÂ² – 2Â²) âŸ¹ (x + 2) (x – 2).

Also apply the identity (aÂ³ + bÂ³) = (a + b) (aÂ² – a b + bÂ²) to factor (xÂ³ + 64).

(xÂ³ + 64) âŸ¹ (xÂ² – 4x + 16)

= [(x + 4) (x â€“ 4)/)/ (x – 2)] * [(x + 2) (x – 2)/ (xÂ² – 4x + 16)]

Cancel common terms to get;

= (x – 4) (x + 2)/ (xÂ² – 4x + 16)

*Example 9*

Simplify [(xÂ² – 9 yÂ²)/ (3 x – 3y)]Â *Â [(xÂ² – yÂ²)/ (xÂ² + 4 x y + 3 yÂ²)]

__Solution__

Apply the algebraic identity (aÂ²-bÂ²) = (a + b) (a – b) to factor (xÂ²- (3y) Â² and (xÂ² – yÂ²)

âŸ¹ (xÂ²-(3y) Â² = (x + 3y) (x-3y)

âŸ¹ (xÂ² – yÂ²) = (x + y) (x – y).

Factor (xÂ² + 4 x y + 3 yÂ²)

=Â xÂ² + 4 x y + 3 yÂ²

= xÂ² + x yÂ + 3 x y + 3 yÂ²

= x (x + y) + 3y (x + y)

= (x + y) (x + 3y)

Cancel common terms to get:

= (x – 3y)/3