9.2: Multiply Rational Expressions
To multiply rational expressions , multiply the numerator expressions and multiply the denominator expressions. Then, if possible, simplify by factoring both the numerator and denominator and removing common factors.
Try to use factor by grouping when working with a \(4\)-term polynomial.
Note: Common denominators are not needed when multiplying rational expressions!
Multiply the rational expressions:
- \(\dfrac{10t}{6t − 12} ∗ \dfrac{20t − 40}{30t^3}\)
- \(\dfrac{7x + 14}{ 2x^2 − 8} ∗ \dfrac{x^2 + 3x − 10}{14x + 21}\)
- \(\dfrac{3x^3 − 24}{2x^2 − 14x + 20} ∗ \dfrac{4x^3 − 20x^2 + 3x − 15}{3x^2 + 6x + 12}\)
Solution
- \(\begin{array} &&\dfrac{10t}{6t − 12} ∗ \dfrac{20t − 40}{30t^3} &\text{Example problem} \\ &\dfrac{10t}{6t − 12} ∗ \dfrac{20(t − 2)}{30t^3} &\text{Factor all numerators and denominators} \\ &\dfrac{(2)(5)(t)(2)(2)(5)(t − 2)}{(2)(3)(t − 2)(2)(3)(5)(t)(t^2)} &\text{Factor numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{(2)}(5)\cancel{(t)}(2)(2)\cancel{(5)}\cancel{(t − 2)}}{\cancel{(2)}(3)\cancel{(t − 2)}(2)(3)\cancel{(5)}\cancel{(t)}(t^2)} &\text{Remove common factors} \\ &\dfrac{(5)(2)(2)}{(3)(2)(3)(t^2)} &\text{Simplify} \\ &\dfrac{20}{18t^2} &\text{Final answer (it is good practice to show final answer as factored as possible)} \end{array} \)
- \(\begin{array} &&\dfrac{7x + 14}{2x^2 − 8} ∗ \dfrac{x^2 + 3x − 10}{14x + 21} &\text{Example problem} \\ &\dfrac{7(x + 2)}{2(x^2 − 4)} ∗ \dfrac{(x + 5)(x − 2)}{7(2x + 3)} &\text{Factor all numerators and denominators} \\ &\dfrac{7(x + 2)(x + 5)(x − 2)}{2(x + 2)(x − 2)(7)(2x + 3)} &\text{Further factor algebraic expressions and numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{7}\cancel{(x + 2)}(x + 5)(x − 2)}{2\cancel{(x + 2)}(x − 2)\cancel{(7)}(2x + 3)} &\text{Remove common factors} \\ &\dfrac{(x + 5)(x − 2)}{2(x − 2)(2x + 3)} &\text{Final answer} \end{array}\)
- \(\begin{array} &&\dfrac{3x^3 − 24}{2x^2 − 14x + 20} ∗ \dfrac{4x^3 − 20x^2 + 3x − 15}{3x^2 + 6x + 12} &\text{Example problem} \\ &\dfrac{3(x^3 − 8)}{2(x^2 − 7x + 10)} ∗ \dfrac{4x^2 (x − 5) + 3(x − 5)}{3(x^2 + 2x + 4)} &\text{Factor all numerators and denominators. Use factor by grouping for the \(4\)-term polynomial.} \\ &\dfrac{3(x − 2)(x^2 + 2x + 4)}{2(x − 5)(x − 2)} ∗ \dfrac{(4x^2 + 3)(x − 5)}{3(x + 2)(x + 2)} &\text{Further factor algebraic expressions} \\ &\dfrac{3(x − 2)(x^2 + 2x + 4)(4x^2 + 3)(x − 5)}{2(x − 5)(x − 2)(3)(x + 2)(x + 2)} &\text{Further factor algebraic expressions and numbers into prime factors and write with one division bar} \\ &\dfrac{\cancel{3}\cancel{(x − 2)}(x^2 + 2x + 4)(4x^2 + 3)\cancel{(x − 5)}}{2\cancel{(x − 5)}\cancel{(x − 2)}\cancel{(3)}(x + 2)(x + 2)} &\text{Remove common factors} \\ &\dfrac{(x^2 + 2x + 4)(4x^2 + 3)}{2(x + 2)(x + 2)} &\text{Final answer } \\ &\dfrac{(x^2 + 2x + 4)(4x^2 + 3)}{2(x + 2)^2} &\text{Final answer can also be written in this form} \end{array}\)
Multiply the rational expressions:
- \(\dfrac{x^2 + 4x + 3}{2x^2 − x − 10} ∗ \dfrac{2x^2 + 4x^3}{x^2 + 3x} ∗ \dfrac{x}{x^2 + 3x + 2}\)
- \(\dfrac{x^2 + 2x − 15}{x^2 − 4x − 45} ∗ \dfrac{x^2 − 5x − 36}{x^2 + x − 12}\)
- \(\dfrac{x^2 + 3x − 40}{x^2 + 2x − 35} ∗ \dfrac{x^2 + 3x − 18}{4x^2 − 5x − 32x + 40}\)