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Multiplication of rational expression
In multiplying rational expressions, we multiply the numerators and denominators, then express the product in its lowest term. This involves: - Multiplying the numerators and denominators of each rational expression - Using cross-cancellation to express the product in lowest terms - Examples are provided to demonstrate multiplying rational expressions and expressing the product in lowest terms through cross-cancellation.
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Multiplication of rational expression
- 2. In multiplying rationalexpressions, we multiply the numerators and denominators, then express the product in its lowest term. The product of rational expressions 𝑎 𝑏 and 𝑐 𝑑 , where a, b, c, and d are polynomials and b ≠ 0 and d ≠ 0 is given by 𝑎 𝑏 . 𝑐 𝑑 = 𝑎𝑐 𝑏𝑑 .
- 3. Example 1. 𝟐 𝟑 . 𝟏 𝟒 Example 2. 𝟏𝟓 𝟐𝒚𝟐. 𝒚𝟒 𝟐𝟎 = (2)(1) (3)(4) = 2 12 (2) = 𝟏 𝟔 Or use cross-cancellation = 2 3 . 1 4 = 1 2 3 . 1 4 2 = (1)(1) (3)(2) = 𝟏 𝟔 = 𝟏𝟓 𝟐𝒚𝟐 . 𝒚𝟒 𝟐𝟎 = 15𝑦4 40𝑦2(5y2) = 𝟑𝒚𝟐 𝟖 By cross-cancellation = 15 2𝑦2 . 𝑦4 20 = 𝟑 15 𝟐 2𝑦2 . 𝑦4 𝒚𝟐 20 𝟒 = 𝟑𝒚𝟐 𝟖
- 4. Example 3. 𝟒𝒂𝟐 𝒃 𝒄𝟑 . 𝟑𝒄𝟓 𝟖𝒂𝟑 𝒃 Example4. 𝒙𝟐 − 𝟒 𝟐𝒙 . 𝟖𝒙𝟐 𝒙+𝟐 = 𝟒𝒂𝟐 𝒃 𝒄𝟑 . 𝟑𝒄𝟓 𝟖𝒂𝟑 𝒃 = 12𝑎2 𝑏𝑐5 8𝑎3 𝑏𝑐3 (4a2bc3) = 𝟑𝒄𝟐 𝟐𝒂 Cross-multiplication = 4𝑎2 𝑏 𝑐3 . 3𝑐5 8𝑎3 𝑏 = 𝟏 4𝑎2 𝑏 𝑐3 . 3𝑐5 𝒄𝟐 8𝑎3 𝑏 𝟐𝒂 = (1)(3)(𝑐2 ) 2𝑎 = 𝟑𝒄𝟐 𝟐𝒂 = 𝑥2 − 4 2𝑥 . 8𝑥2 𝑥+2 = (𝑥+2)(𝑥 −2) 2𝑥 . (2𝑥)(4𝑥) 1(𝑥+2) = (𝑥 −2)(4𝑥) 1 = (x – 2)(4x)
- 5. Example 5. 𝟐𝒙 −𝟔 𝟑𝒙 . 𝒙 𝒙−𝟑 Example 6. 𝒙+𝟓 𝟒𝒙 . 𝟏𝟐𝒙𝟐 𝒙𝟐 +𝟕𝒙+𝟏𝟎 = 2𝑥 −6 3𝑥 . 𝑥 𝑥 −3 = 2 (𝑥 −3) 3𝑥 . 𝑥 1(𝑥 −3) = 2 (3)(1) = 𝟐 𝟑 = 𝒙+𝟓 𝟒𝒙 . 𝟏𝟐𝒙𝟐 𝒙𝟐 +𝟕𝒙+𝟏𝟎 = 1(𝑥+5) 4𝑥 . (4𝑥)(3𝑥) (𝑥+5)(𝑥+2) = (1)(3𝑥) (𝑥+2) = 𝟑𝒙 𝒙+𝟐