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MULTIPLYING & DIVIDING RATIONAL
ALGEBRAIC EXPRESSIONS
MODULE 7
HOW TO MULTIPLY
RATIONAL EXPRESSIONS?
• Operations like multiplication and
division are very common in
fractions.
• These operations can also be
performed to rational expressions.
• To obtain the product of two or
more rational expressions, we get
the product of the numerators as
well as the product of the
denominators of the given
expressions.
• Rational expressions are multiplied
in a similar manner as multiplying
rational numbers.
• The rational expressions must be
factored first before multiplying
them so that common factors can
be cancelled.
Multiply & simplify:
EXAMPLE
1.
𝟐𝒙𝒚 𝟐
𝟑𝒛 𝟑 ∙
𝟏𝟓𝒛 𝟓
𝟒𝒙 𝟐 𝒚
=
𝟐 ∙ 𝟑 ∙ 𝟓 ∙ 𝒙 ∙ 𝒚 ∙ 𝒚 ∙ 𝒛 ∙ 𝒛 ∙ 𝒛 ∙ 𝒛 ∙ 𝒛
𝟑 ∙ 𝟐 ∙ 𝟐 ∙ 𝒙 ∙ 𝒙 ∙ 𝒚 ∙ 𝒛 ∙ 𝒛 ∙ 𝒛
=
𝟓𝒚𝒛 𝟐
𝟐𝒙
Multiply & simplify:
EXAMPLE
2.
𝒙
𝒙−𝟏
∙
𝒙−𝟏 𝟐
𝟑𝒙 𝟐 ∙
𝟗
𝒙 𝟐−𝒙
=
𝒙 ∙ 𝒙 − 𝟏 𝒙 − 𝟏 ∙ 𝟑 ∙ 𝟑
𝒙 − 𝟏 ∙ 𝟑 ∙ 𝒙 ∙ 𝒙 ∙ 𝒙 𝒙 − 𝟏
=
𝟑
𝒙 𝟐
Multiply & simplify:
EXAMPLE
3.
𝒂 𝟐 𝒃−𝟐𝒂𝒃 𝟐
𝒂 𝟐−𝟒𝒂𝒃+𝟒𝒃 𝟐 ∙
𝒂 𝟐−𝟒𝒃 𝟐
𝒂𝒃
=
𝒂𝒃 𝒂 − 𝟐𝒃 ∙ 𝒂 + 𝟐𝒃 𝒂 − 𝟐𝒃
𝒂 − 𝟐𝒃 𝒂 − 𝟐𝒃 ∙ 𝒂𝒃
=
𝒂 + 𝟐𝒃
𝟏
= 𝒂 + 𝟐𝒃
HOW TO DIVIDE RATIONAL
EXPRESSIONS?
• In dividing rational expressions, we
multiply the numerator by the
reciprocal of the denominator.
•
𝐴
𝐵
÷
𝐶
𝐷
=
𝐴
𝐵
∙
𝐷
𝐶
=
𝐴𝐷
𝐵𝐶
Divide & simplify:
EXAMPLE
1.
𝒙 𝟐+𝟐𝒙−𝟖
𝒙 𝟐+𝟒𝒙+𝟑
÷
𝒙−𝟐
𝟑𝒙+𝟑
=
𝒙 𝟐
+ 𝟐𝒙 − 𝟖
𝒙 𝟐 + 𝟒𝒙 + 𝟑
∙
𝟑𝒙 + 𝟑
𝒙 − 𝟐
=
𝒙 + 𝟒 𝒙 − 𝟐 ∙ 𝟑 𝒙 + 𝟏
𝒙 + 𝟏 𝒙 + 𝟑 ∙ (𝒙 − 𝟐)
=
𝟑 𝒙 + 𝟒
𝒙 + 𝟑
𝒐𝒓
𝟑𝒙 + 𝟏𝟐
𝒙 + 𝟑
Divide & simplify:
EXAMPLE
2.
𝒙 𝟐−𝟔𝒙+𝟗
𝟒𝒙−𝟏𝟐
÷ 𝒙 − 𝟑 =
𝒙 𝟐
− 𝟔𝒙 + 𝟗
𝟒𝒙 − 𝟏𝟐
∙
𝟏
𝒙 − 𝟑
=
𝒙 − 𝟑 𝒙 − 𝟑 ∙ 𝟏
𝟒 𝒙 − 𝟑 ∙ (𝒙 − 𝟑)
=
𝟏
𝟒
Divide & simplify:
EXAMPLE
3. 𝒙 𝟐 − 𝟑𝒙 − 𝟏𝟎 ÷
𝒙 𝟐+𝟑𝒙−𝟏𝟎
𝒙 𝟐−𝟐𝟓
=
𝒙 𝟐−𝟑𝒙−𝟏𝟎
𝟏
∙
𝒙 𝟐−𝟐𝟓
𝒙 𝟐+𝟑𝒙−𝟏𝟎
=
𝒙+𝟐 𝒙−𝟓 ∙ 𝒙+𝟓 𝒙−𝟓
𝟏∙ 𝒙−𝟐 𝒙+𝟓
=
𝒙+𝟐 𝒙−𝟓 𝒙−𝟓
𝟏∙ 𝒙−𝟐
=
𝒙+𝟐 𝒙−𝟓 𝟐
𝒙−𝟐
THANK YOU

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Multiplying & dividing rational algebraic expressions

  • 1. MULTIPLYING & DIVIDING RATIONAL ALGEBRAIC EXPRESSIONS MODULE 7
  • 3. • Operations like multiplication and division are very common in fractions. • These operations can also be performed to rational expressions.
  • 4. • To obtain the product of two or more rational expressions, we get the product of the numerators as well as the product of the denominators of the given expressions.
  • 5. • Rational expressions are multiplied in a similar manner as multiplying rational numbers. • The rational expressions must be factored first before multiplying them so that common factors can be cancelled.
  • 6. Multiply & simplify: EXAMPLE 1. 𝟐𝒙𝒚 𝟐 𝟑𝒛 𝟑 ∙ 𝟏𝟓𝒛 𝟓 𝟒𝒙 𝟐 𝒚 = 𝟐 ∙ 𝟑 ∙ 𝟓 ∙ 𝒙 ∙ 𝒚 ∙ 𝒚 ∙ 𝒛 ∙ 𝒛 ∙ 𝒛 ∙ 𝒛 ∙ 𝒛 𝟑 ∙ 𝟐 ∙ 𝟐 ∙ 𝒙 ∙ 𝒙 ∙ 𝒚 ∙ 𝒛 ∙ 𝒛 ∙ 𝒛 = 𝟓𝒚𝒛 𝟐 𝟐𝒙
  • 7. Multiply & simplify: EXAMPLE 2. 𝒙 𝒙−𝟏 ∙ 𝒙−𝟏 𝟐 𝟑𝒙 𝟐 ∙ 𝟗 𝒙 𝟐−𝒙 = 𝒙 ∙ 𝒙 − 𝟏 𝒙 − 𝟏 ∙ 𝟑 ∙ 𝟑 𝒙 − 𝟏 ∙ 𝟑 ∙ 𝒙 ∙ 𝒙 ∙ 𝒙 𝒙 − 𝟏 = 𝟑 𝒙 𝟐
  • 8. Multiply & simplify: EXAMPLE 3. 𝒂 𝟐 𝒃−𝟐𝒂𝒃 𝟐 𝒂 𝟐−𝟒𝒂𝒃+𝟒𝒃 𝟐 ∙ 𝒂 𝟐−𝟒𝒃 𝟐 𝒂𝒃 = 𝒂𝒃 𝒂 − 𝟐𝒃 ∙ 𝒂 + 𝟐𝒃 𝒂 − 𝟐𝒃 𝒂 − 𝟐𝒃 𝒂 − 𝟐𝒃 ∙ 𝒂𝒃 = 𝒂 + 𝟐𝒃 𝟏 = 𝒂 + 𝟐𝒃
  • 9. HOW TO DIVIDE RATIONAL EXPRESSIONS?
  • 10. • In dividing rational expressions, we multiply the numerator by the reciprocal of the denominator. • 𝐴 𝐵 ÷ 𝐶 𝐷 = 𝐴 𝐵 ∙ 𝐷 𝐶 = 𝐴𝐷 𝐵𝐶
  • 11. Divide & simplify: EXAMPLE 1. 𝒙 𝟐+𝟐𝒙−𝟖 𝒙 𝟐+𝟒𝒙+𝟑 ÷ 𝒙−𝟐 𝟑𝒙+𝟑 = 𝒙 𝟐 + 𝟐𝒙 − 𝟖 𝒙 𝟐 + 𝟒𝒙 + 𝟑 ∙ 𝟑𝒙 + 𝟑 𝒙 − 𝟐 = 𝒙 + 𝟒 𝒙 − 𝟐 ∙ 𝟑 𝒙 + 𝟏 𝒙 + 𝟏 𝒙 + 𝟑 ∙ (𝒙 − 𝟐) = 𝟑 𝒙 + 𝟒 𝒙 + 𝟑 𝒐𝒓 𝟑𝒙 + 𝟏𝟐 𝒙 + 𝟑
  • 12. Divide & simplify: EXAMPLE 2. 𝒙 𝟐−𝟔𝒙+𝟗 𝟒𝒙−𝟏𝟐 ÷ 𝒙 − 𝟑 = 𝒙 𝟐 − 𝟔𝒙 + 𝟗 𝟒𝒙 − 𝟏𝟐 ∙ 𝟏 𝒙 − 𝟑 = 𝒙 − 𝟑 𝒙 − 𝟑 ∙ 𝟏 𝟒 𝒙 − 𝟑 ∙ (𝒙 − 𝟑) = 𝟏 𝟒
  • 13. Divide & simplify: EXAMPLE 3. 𝒙 𝟐 − 𝟑𝒙 − 𝟏𝟎 ÷ 𝒙 𝟐+𝟑𝒙−𝟏𝟎 𝒙 𝟐−𝟐𝟓 = 𝒙 𝟐−𝟑𝒙−𝟏𝟎 𝟏 ∙ 𝒙 𝟐−𝟐𝟓 𝒙 𝟐+𝟑𝒙−𝟏𝟎 = 𝒙+𝟐 𝒙−𝟓 ∙ 𝒙+𝟓 𝒙−𝟓 𝟏∙ 𝒙−𝟐 𝒙+𝟓 = 𝒙+𝟐 𝒙−𝟓 𝒙−𝟓 𝟏∙ 𝒙−𝟐 = 𝒙+𝟐 𝒙−𝟓 𝟐 𝒙−𝟐