## MULTIPLYING RATIONAL EXPRESSIONS

How to multiply rational expressions :

1. Multiply numerators and denominators.

2. Factor the numerator and denominator.

3. Cancel common factors to simplify.

Multiply the rational expressions and show the product in simplest form :

Example 1 :

[(x2 - 2x)/(2x + 4)]  [(3x + 6)/(5x - 10)]

Solution :

=  [(x2 - 2x)/(2x + 4)] ⋅ [(3x + 6)/(5x - 10)]

Multiply numerators and denominators.

=  [(x2 - 2x)(3x + 6)] /[(2x + 4)(5x - 10)]

Factor the numerator and denominator.

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

=  [3x(x - 2)(x + 2)] /[10(x + 2)(x - 2)]

Cancel common factors.

=  3x/10

Example 2 :

[(6x + 6)/(x2 - 4)] ⋅ [(x2 + 6x + 8)/(3x + 3)]

Solution :

=  [(6x + 6)/(x2 - 4)] ⋅ [(x2 + 6x + 8)/(3x + 3)]

Multiply numerators and denominators.

=  [(6x + 6)(x2 + 6x + 8)] / [(x2 - 4)(3x + 3)]

Factor the numerator and denominator.

=  [6(x + 1)(x + 2)(x + 4)] / [(x + 4)(x - 4)3(x + 1)]

=  [6(x + 1)(x + 2)(x + 4)] / [3(x + 4)(x - 4)(x + 1)]

Cancel common factors.

=  [6(x + 2)] / [3(x - 4)]

=  2(x + 2)/(x - 4)

Example 3 :

[(a + b)/(a - b)] ⋅ [(a3 - b3)/(a3 + b3)]

Solution :

=  [(a + b)/(a - b)] ⋅ [(a3 - b3)/(a3 + b3)]

Multiply numerators and denominators.

=  [(a + b)(a3 - b3)]/[(a - b)(a3 + b3)]

Factor the numerator and denominator.

=  [(a + b)(a - b)(a2 + ab + b2)]/[(a - b)(a + b)(a2 - ab + b2)]

Cancel common factors.

=  (a2 + ab + b2) / (a2 - ab + b2)

Example 4 :

Solution :

=  (5ab/15cd) ⋅ (4bc/32ad) ⋅ (16ac/2bc)

Multiply numerators and denominators.

=  (5ab ⋅ 4bc ⋅ 16ac) / (15cd ⋅ 32ad ⋅ 2bc)

=  320a2b2c/ 960abc2d2

Cancel common factors.

=  ab/3d2

Example 5 :

[(x2 - 2x + 1)/(x2 - 3x + 2)] ⋅ [(3x - 6)/(6x - 6)]

Solution :

=  [(x2 - 2x + 1)/(x2 - 3x + 2)] ⋅ [(3x - 6)/(6x - 6)]

Multiply numerators and denominators.

[(x2 - 2x + 1)(3x - 6)] / [(x2 - 3x + 2)(6x - 6)]

Factor the numerator and denominator.

=  [(x - 1)(x - 1)3(x - 2)] / [(x - 2)(x - 1)6(x - 1)]

=  [3(x - 1)(x - 1)(x - 2)] / [6(x - 2)(x - 1)(x - 1)]

Cancel common factors.

=  3/6

=  1/2

Example 6 :

[(x2 - 25)/(x + 3)] ⋅ [(x2 - 9)/(x + 5)2]

Solution :

=  [(x2 - 25)/(x + 3)] ⋅ [(x2 - 9)/(x + 5)2]

Multiply numerators and denominators.

=  [(x2 - 25)(x2 - 9)] ⋅ [(x + 3)(x + 5)2]

Factor the numerator and denominator.

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

Cancel common factors.

=  [(x - 5)(x - 3)] / (x + 5)

Example 7 :

[(x2 - 9y2)/(3x - 3y)] ⋅ [(x - y)/(x + 3y)]

Solution :

=  [(x2 - 9y2)/(3x - 3y)] ⋅ [(x - y)/(x + 3y)]

Multiply numerators and denominators.

=  [(x2 - 9y2)(x - y)] / [(3x - 3y)(x + 3y)]

Factor the numerator and denominator.

=  [(x + 3y)(x - 3y)(x - y)] / [3(x - y)(x + 3y)]

Cancel common factors.

=  (x - 3y)/3

Example 8 :

[(x2 - 16)/(x - 2)] ⋅ [(x - 2)/(x3 + 64)]

Solution :

=  [(x2 - 16)/(x - 2)] ⋅ [(x - 2)/(x3 + 64)]

Multiply numerators and denominators.

=  [(x2 - 16)(x - 2)] / [(x - 2)(x3 + 64)]

=  [(x2 - 42)(x - 2)] / [(x - 2)(x3 + 43)]

Factor the numerator and denominator.

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

Cancel common factors.

=  (x - 4)/(x2 - 4x + 16)

Example 9 :

[(p2 - 1)/p] ⋅ [p3/(p - 1)] ⋅ [1/(p + 1)]

Solution :

=  [(p2 - 1)/p] ⋅ [p3/(p - 1)] ⋅ [1/(p + 1)]

Multiply numerators and denominators.

=  [(p2 - 1) ⋅ p3 ⋅ 1] / [p(p - 1)(p + 1)]

=  p3(p2 - 1) / p(p2 - 12)

=  p3(p2 - 1) / p(p2 - 1)

Cancel common factors.

=  p2

Example 10 :

[(px - 2p)/(qx - 3q)] ⋅ [(bx - 3b)/(ax - 2a)]

Solution :

=  [(px - 2p)/(qx - 3q)] ⋅ [(bx - 3b)/(ax - 2a)]

Multiply numerators and denominators.

=  [(px - 2p)(bx - 3b)] / [(qx - 3q)(ax - 2a)]

Factor the numerator and denominator.

=  [p(x - 2)b(x - 3)] / [q(x - 3)a(x - 2)]

=  [bp(x - 2)(x - 3)] / [aq(x - 3)(x - 2)]

Cancel common factors.

=  bp/aq

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