Which of the following is a polynomial with roots 4, - 5, and 7?

If x=a is a zero of f (x) then (x-a) is a factor of f (x). Then, since x = 4,6,-7 are all zeros of f (x), we may

write f (x) = (x-4) (x-6) (x+7) = (x-4) (x^2 + x - 42) = x^3 - 3x^3 - 46x + 168. This is choice B. I have no

difficulty in expanding and mentally collecting terms to write the coefficients of descending powers of x in 1 step as I proceed. This is a very useful technique. It forces a systematic, concentrated method on one which prevents sloppy mistakes.

Roots means that x = 4 x = - 5 and x=7. If you're familiar with finding roots of a polynomial, you would know that you factor the polynomial, then you take each factor and set it equal to 0 and you solve for x. well now youre just doing the reverse. take each of the roots, turn them into factors, put them together and multiply them out:

x=4 x = - 5 x=7

(x - 4) (x + 5) (x - 7)

(x - 4) (x + 5) (x - 7) now multiply it out

(x² - 4x + 5x + 28) (x - 7)

(x² + x + 28) (x - 7)

x³ - 7x² + x² - 7x + 28x - 196

x³ - 6x² + 14x - 196

f (x) = x³ - 6x² + 14x - 196 there's your answer