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How Do You Multiply a Rational Expression by a Polynomial?

Multiply and simplify the following: [(4x+8)/(x2-25)]•(x-5)

Summary

  1. We want to represent the x-5 as a fraction so we can multiply the numerator and denominators together
  2. Do this by just putting x-5 over 1
  3. We need to factor and then cancel out anything that we can
  4. Factor a 4 out of 4x+8
  5. x2 is a difference of squares, so that factors in to (x-5)(x+5)
  6. The only thing that cancels out is the x-5 in the numerator and denominator
  7. We can multiply all the factors back together to find our answer

Notes

    1. This will help us multiply these fractions together later
    1. Anything can be rewritten as a fraction by just putting it over 1
    1. In our case, this means multiplying 4x+8 and x-5
    2. Also, we can multiply x2-25 by 1
    1. We're multiplying together our rational expressions
    2. Don't multiply out (4x+8)(x-5) yet because we'll want to cancel out any common factors first
    1. Now we have to factor out anything in the numerator and denominator that we can
    1. The numerator is (4x+8)(x-5)
    1. We've already factored a 4 out of 4x+8, but the other part of the numerator, x-5, can't be factored
    1. x2 and 25 are both perfect squares and they are being subtracted, so we have a difference of squares
    2. That means we can factor easily by taking the square root of each term and using those as the terms in our factors
    3. The square root of x2 is x and the square root of 25 is 5
    4. So x2-25 factors into (x-5)(x+5)
    1. A common factor in this case is a factor that is in both the numerator and denominator of our rational expression
    1. The factors in the numerator are 4, x+2, and x-5
    2. Do you see anything in the denominator that is also a factor in the numerator?
    1. The denominator has the factors x-5 and x+5
    2. The 4 cannot be canceled with the x-5 or x+5
    1. The denominator has the factors x-5 and x+5
    2. x+2 cannot be canceled with either x-5 or x+5
    1. The denominator has the factors x-5 and x+5
    1. Now we have to take each factor and recombine them with other factors
    1. After canceling the x-5, we were left with 4(x+2) in the numerator
    2. Multiply 4•x to get 4x and 4•2 to get 8
    1. Anything can be rewritten as a fraction by just putting it over 1
    2. Common factors are factors that are in both the numerator and denominator of a fraction