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# Multiplying Rational Expressions – Techniques & Examples

To **learn how to multiply rational expressions**, let’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 × c/d = a × c/b × d. Let’s take a look at the examples below:

- Multiply 2/7 by 3/5

__Solution__

2/7 × 3/5

= 2 × 3/7 × 5= 6/35

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

__Solution__

5/9 × (-3/4)

= 5 × -3/9 × 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:

- (a² – b²) = (a + b) (a – b)
- (x² – 4²) = (x + 4) (x – 4)
- (x² – 2²) = (x + 2) (x – 2)
- (a³ + b³) = (a + b) (a² – a b + b²)

*Example 1*

Simplify (x² – 2x) / (x + 2) * (3 x + 6)/ (x – 2)

__Solution__

Factor the numerators,

(x² – 2x) / (x + 2) * (3 x + 6)/ (x – 2)

⟹ x (x – 2) / (x + 2) * 3(x + 2)/ (x – 2)

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

⟹ 3x

*Example 2*

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

__Solution__

First, factor the numerators and denominators of both fractions.

[(x – 4) (x + 1)/ (x + 1) (x – 2)] * [(x + 2) (x – 2)/ (x – 4) (x + 5)]

Cancel out common terms and rewrite the remaining terms

= x + 2/x + 5

*Example 3*

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

__Solution__

Factor the rational expressions.

⟹ [-4x (x – 3)/ (x – 3) (x + 4)] * [(x – 2) (x + 4)/x (x + 2) (x – 2)]

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

= -4/x + 2

*Example 4*

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

__Solution__

Factor the fractions

⟹ [(2x – 3) (x + 2)/ (3x + 1) (x – 3)] * [(x – 30(x – 4)/ (2x + 1) (x – 4)]

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

⟹ [(2x – 3) (x + 2)/ (3x + 1) (2x + 1)]

*Example 5*

Simplify [(x² – 81)/ (x² – 4)] * [(x² + 6 x + 8)/ (x² – 5 x – 36)]

__Solution__

Factor the numerators and denominators of each fraction.

⟹ [(x + 9) (x – 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 [(x² – 3 x – 10)/ (x² – x – 20)] * [(x² – 2 x + 4)/ (x³ + 8)]

__Solution__

Factor out (x³ + 8) using the algebraic identity (a³ + b³) = (a + b) (a² – a b + b²).

⟹ (x³ + 8) = (x + 2) (x² – 2 x + 4).

⟹ (x² – 3 x – 10) = (x – 5) (x + 2)

⟹ (x² – x – 20) = (x – 5) (x + 4)

[(x² – 3 x – 10)/ (x² – x – 20)] * [(x² – 2 x + 4)/ (x³ + 8)] = [(x – 5) (x + 2)/ (x – 5) (x + 4)] * [(x² – 2 x + 4)/ (x + 2) (x² – 2 x + 4)]

Now, cancel out common terms to get;

= 1/ (x + 4).

*Example 7*

Simplify [(x + 7)/ (x² + 14 x + 49)] * [(x² + 8x + 7)/ (x + 1)]

__Solution__

Factor the fractions.

⟹ (x² + 14 x + 49) = (x + 7) (x + 7)

⟹ (x² + 8x + 7) = (x + 1) (x + 7)

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

On cancelling common terms, we get the answer as;

= 1

*Example 8*

Multiply [(x² – 16)/ (x – 2)] * [(x² – 4)/ (x³ + 64)]

__Solution__

Use the algebraic identity (a² – b²) = (a + b) (a – b) to factor (x² – 16) and (x² – 4).

(x² – 4²) ⟹ (x + 4) (x – 4)

(x² – 2²) ⟹ (x + 2) (x – 2).

Also apply the identity (a³ + b³) = (a + b) (a² – a b + b²) to factor (x³ + 64).

(x³ + 64) ⟹ (x² – 4x + 16)

= [(x + 4) (x – 4)/)/ (x – 2)] * [(x + 2) (x – 2)/ (x² – 4x + 16)]

Cancel common terms to get;

= (x – 4) (x + 2)/ (x² – 4x + 16)

*Example 9*

Simplify [(x² – 9 y²)/ (3 x – 3y)] * [(x² – y²)/ (x² + 4 x y + 3 y²)]

__Solution__

Apply the algebraic identity (a²-b²) = (a + b) (a – b) to factor (x²- (3y) ² and (x² – y²)

⟹ (x²-(3y) ² = (x + 3y) (x-3y)

⟹ (x² – y²) = (x + y) (x – y).

Factor (x² + 4 x y + 3 y²)

= x² + 4 x y + 3 y²

= x² + x y + 3 x y + 3 y²

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

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

Cancel common terms to get:

= (x – 3y)/3

*Practice Questions*

Simplify the following rational expressions:

- [(x² – 16)/ (x² – 3x + 2)] * [(x²-4)/ (x³ + 64)] * [(x² – 4x + 16)/ (x² – 2x – 8)]
- [(a + b)/ (a – b)] * [(a³ – b³)/ (a³ + b³)]
- [(x² – 4x – 12)/ (x² – 3x – 18)] * [(x² – 2x – 3)/ (x² + 3 x + 2)]
- [(p² – 1)/p] x [p²/ (p – 1)] x [1/ (p + 1)]
- [(2 x – 1)/ (x² + 2x + 4)] * [(x⁴ – 8 x)/ (2 x² + 5 x -3)] * [(x + 3)/ (x²- 2x)]
- [(x² – 16)/ (x² – 3x + 2)]
*****[(x² – 4)/(x³ + 64)]*****[(x² – 4x + 16)/ (x² – 2x – 8)] - [(x
^{2}– 8x = 12)/(x^{2}– 16)] * [(4x + 16) (x^{2 }– 4x + 4)]

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