Dividing polynomials? please help! ?

x² - 18x + 80 / x - 10

first, multiply the divisor (x - 10) by something that will make its first term match the first term of the dividend (x² - 18x + 80), in this case that's x

x (x - 10) = x² - 10x

now subtract this from the dividend by combining like terms:

(x² - 18x + 80) - (x² - 10x) = -8x + 80

so the first term of your quotient is x

now repeat the process by multiplying the divisor by -8 (to make the first terms of divisor and what's left of the dividend match):

-8(x - 10) = -8x + 80

then subtract that from what remains of the dividend:

(-8x + 80) - (-8x + 80) = 0

so the second term of your quotient is -8 and the answer is (x - 8)

You can check this by multiplying (x - 8)(x - 10) = x² - 18x + 80 so that's the correct answer. That's really what the question is: what quantity, when multiplied by (x - 10) gives the result (x² - 18x + 80)?

This is actually easier to see and understand if you lay it out like a long division problem.

(x^2 - 18x + 80)/(x - 10)
= (x^2 - 8x - 10x + 80)/(x - 10)
= (x - 8)(x - 10)/(x - 10) (cancel out x - 10)
= (x - 8)/1
= x - 8

(x-10)(x-8)/(x-10)
=x - 8