Multiply - Rational Expressions
Rational Expressions - Multiply - How it Works - Video
Example 1
Example 1:
Here we have the expression (x2 - 3x - 4) / (x2 - 4x - 5) . The best bet when dealing with rational expressions is to factor them and see if we can cancel any factors out so we can make it smaller.
Factor: x2 - 3x - 4
Factor: x2 - 4x - 5
So x2 - 3x - 4 becomes (x + 1)(x - 4).
So x2 - 4x - 5 becomes (x + 1)(x - 5).
Remember these questions for the most part will cancel so if you find one factor in the numerator, it will probably be in the denominator. So you can use that to help you to find other factors.
(x2 - 3x - 4) / (x2 - 4x - 5)
(x - 4)(x + 1) / (x - 5)(x + 1)
(x - 4) * ̶(̶x̶ ̶+̶ ̶1̶)̶ / (x - 5) * ̶(̶x̶ ̶+̶ ̶1̶)̶
(x - 4) / (x - 5)
Given
Factored each polynomial
Canceled out the factors (x + 1) in the numerator and denominator
Wrote the answer
Example 2
Example 2:
Here we have the expression (x2 - x - 6) / (x2 - 3x - 10) * (x2 + 6x - 7) / (x2 + 5x - 14). The best bet when dealing with rational expressions is to factor them and see if we can cancel any factors out so we can make it smaller.
Factor: x2 - x - 6
Factor: x2 - 3x - 10
Factor: x2 + 6x - 7
Factor: x2 + 5x - 14
So x2 - x - 6 becomes (x + 2)(x - 3).
So x2 - 3x - 10 becomes (x + 2)(x - 5).
So x2 + 6x - 7 becomes (x - 1)(x + 7).
So x2 + 5x - 14 becomes (x - 2)(x + 7).
Remember these questions for the most part will cancel so if you find one factor in the numerator, it will probably be in the denominator. So you can use that to help you to find other factors.
(x2 - x - 6) / (x2 - 3x - 10) * (x2 + 6x - 7) / (x2 + 5x - 14)
(x + 2)(x - 3) / (x - 5)(x - 2) * (x - 1)(x + 7) / (x + 7)(x - 2)
(x + 2)(x - 3)(x - 1)(x + 7) / (x - 5)(x - 2)(x + 7)(x - 2)
(x - 3) * (x - 1) * (x + 2) * (x + 7) / (x - 5)* (x - 1) * (x + 2) * (x + 7)
(x - 3) * (x - 1) * ̶(̶x̶ ̶+̶ ̶2̶)̶ * ̶(̶x̶ ̶+̶ ̶7̶)̶ / (x - 5) * (x - 1) * ̶(̶x̶ ̶+̶ ̶2̶)̶ * ̶(̶x̶ ̶+̶ ̶7̶)̶
(x - 3)(x - 1) / (x - 5)(x - 2)
Given
Factored each polynomial
Smash the numerators and denominators together
Moved the factors around so common factors are closer
Canceled out the factors (x + 2) and (x + 7) on the top and bottom
Wrote the answer
Example 3
Example 3:
Here we have the expression (x2 + 5x + 6) / (4x2 - 4x - 24) * (2x2 + 8x - 10) / (x2 + 10x + 21). The best bet when dealing with rational expressions is to factor them and see if we can cancel any factors out so we can make it smaller.
Factor: x2 + 5x + 6
Factor: 4x2 - 4x - 24
Factor: 2x2 + 8x - 10
Factor: x2 + 10x + 21
So x2 + 5x + 6 becomes (x + 2)(x + 3).
So 4x2 - 4x - 24 becomes 4(x + 2)(x - 3).
So 2x2 + 8x - 10 becomes 2(x - 1)(x + 5).
So x2 + 10x + 21 becomes (x + 3)(x + 7).
Remember these questions for the most part will cancel so if you find one factor in the numerator, it will probably be in the denominator. So you can use that to help you to find other factors.
(x2 + 5x + 6) / (4x2 - 4x - 24) * (2x2 + 8x - 10) / (x2 + 10x + 21)
(x + 2)(x + 3) / [ 4(x - 5)(x - 2) ] * [ 2(x - 1)(x + 5) ] / (x + 7)(x + 3)
2 (x + 2)(x + 3)(x - 1)(x + 5) / 4(x - 3)(x + 2)(x + 7)(x + 3)
2 * (x - 1) * (x + 2) * (x + 3) * (x + 5) / 4 * (x - 3) * (x + 2) * (x + 3) * (x + 7)
̶2̶ * (x - 1) * ̶(̶x̶ ̶+̶ ̶2̶)̶ * ̶(̶x̶ ̶+̶ ̶3̶)̶ * (x + 5) / ̶4̶ * (x - 3) * ̶(̶x̶ ̶+̶ ̶2̶)̶ * ̶(̶x̶ ̶+̶ ̶3̶)̶ * (x + 7)
(x - 1)(x + 5) / [ 2(x - 3)(x + 7) ]
Given
Factored each polynomial
Smash the numerators and denominators together
Moved the factors around so common factors are closer
Canceled out the factors (x + 2) and (x + 3) on the top and bottom
Wrote the answer