Math A-100 IUSB Sec. 7.2 Detailed notes – Week 12 Multiplying & Dividing Rational Polynomials Just like regular fractions, you can multiply and divide rational fractions. Multiplying Example 1: x2 + 10x + 21_ * x2 - 12x + 32 x 2 + 9x - 52 x2 - 12x - 45 Unlike regular fractions, this can pretty nasty if you try to multiply the tops and then the bottoms, then reduce. x2 + 10x + 21_ * x2 - 12x + 32 x 2 + 9x - 52 = x2 - 12x – 45 x4 + 19x3 – 121x2 – 331x - 1092 x4 – 24x3 +131x2 + 416x - 1440 What on earth do you do this this?!?!?!?!?!? Recall that you can also cancel out parts of the fractions before you multiply. To do this, we need to factor all four parts to see what, if anything, we can cancel. x2 + 10x + 21_ * x2 - 12x + 32 x 2 + 9x - 52 x2 - 12x - 45 becomes (x + 3)(x + 7)_ * (x – 8)(x – 4) (x – 4)(x + 13) (x – 15)(x + 3) NOW cancel (x + 3)(x + 7)_ * (x – 8)(x – 4) (x – 4)(x + 13) (x – 15)(x + 3) Leaving you with (x + 7)_ * (x – 8) (x + 13) (x – 15) We will write the answer in “factored form” (see above) to show that we cannot simply it further Answer: Example 2: (x + 7)(x + 13) (x – 8)(x – 15) x2 – 36 _ * x2 - 1x – 30 x 2 + 7x + 10 (x + 2)2 becomes (x + 6)(x - 6)_ * (x – 6)(x + 5) (x + 2)(x + 5) (x + 2)(x + 2) cancelling (x + 6)(x - 6)_ * (x – 6)(x + 5) (x + 2)(x + 5) (x + 2)(x + 2) Leaving you with (x + 6)_ or just x + 6 (x + 2) x+2 NOTE: you can cancel both from one fraction to the other AND within the same fraction. You are simply reducing the fraction, like 6/8  3/4 Dividing We divide rational polynomials the same way we divide regular fraction: Flip the second fraction and multiply instead! Example 1: x2 + 10x - 24_ divided by x 2 + 9x + 20 x2 - 4x - 12 x2 - 12x + 36 flip & multiply: x2 + 10x - 24_ * x2 - 4x - 12 becomes (x + 2)(x – 12) * (x - 6)(x - 6) (x - 6)(x + 2) (x + 4)(x + 5) Leaving you with (x -12)(x - 6) (x + 4)(x + 5) x 2 – 12x + 36 x2 + 9x + 20 NOTE: Notice that I left the answers above with the ( ) still in them. This is called factored form. We do this because it shows clearly that the fraction cannot be simplified any further. If you were to multiply out the top and bottom, you would get x2 – 18x + 72 x2 + 9x + 20 The only way to prove that this fraction does not reduce is to factor it again, so we leave it in factored form.