How do you multiply #(56+11x-16x^2)*10/(15x^2-11x-56)# and state the excluded values?

Question

Abigail Allen · Accepted Answer

#(56+11x-16x^2)*10/(15x^2-11x-56)#

#= - {160x^2 -110 x - 560}/{15x^2 -11 x - 56}#

excluding #x = -8/5 and x=7/3# There's a #15x^2# in the denominator and a #-16x^2# in the numerator, so the cancelling that we might hope for at first glance doesn't materialize. The excluded values occur at the zeros of the denominator; let's try to factor. We seek a pair of factors of #15# and a pair of factors of #-56# whose sum of products is #-11.# That's a bit of a search, we have #15=1 imes 15= 3 imes 5# # 56=1 imes 56= 2 imes 28 = 4 imes 14 = 7 imes 8# with a minus sign in there for #-56.# Eventually we find #-7 imes 8=-56,# #5 times 3 = 15,# #-7(5)+3(8)=-11 quad sqrt# # 15x^2 -11 x - 56 = (5 x + 8) (3 x - 7) # We could also have used the quadratic formula to find the zeros of the denominator, which are called poles. #x = -8/5 or x=7/3# The multiplication itself is rather vacuous due to the lack of cancelling. #(56+11x-16x^2)*10/(15x^2-11x-56)# #= - {160x^2 -110 x - 560}/{15x^2 -11 x - 56}#

Abigail Allen · Answer

undefined To multiply (56+11x-16x^2)*10/(15x^2-11x-56), you can first simplify the expression by factoring the numerator and denominator. The factored form of the numerator is 10(8-x)(7+2x), and the factored form of the denominator is (5x+7)(3x-8).

Next, you can cancel out any common factors between the numerator and denominator. In this case, there are no common factors to cancel out.

After canceling out common factors, you can multiply the remaining terms in the numerator and denominator.

The excluded values are the values of x that would make the denominator equal to zero. In this case, the excluded values are x = -7/5 and x = 8/3.