Question

Which of the following is a polynomial with roots 5, −6, and 7?
A. f(x) = x3 − 6x2 − 30x + 37
B. f(x) = x3 − 30x2 − 37x + 42
C. f(x) = x3 − 6x2 − 37x + 210
D. f(x) = x3 − 30x2 − 42x + 210

Answer 1

I don't know the answer and I forgot how to do these, but someone on yahoo had a similar question with different numbers. Maybe the answer they got can help you.

https://answers.yahoo.com/question/index?qid=20151029103535AA5FiWd


If you consider the expansion of (x-4)(x-6)(x+7), you see that the constant term must be
.. (-4)(-6)(+7) = 168
This rules out choices A and C.
If you consider the coefficient of the x^2 term, you realize it must be (-4) + (-6) + (7) = -3.
This rules out choice D and suggests choice B.
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If you choose to evaluate the polynomial at each of the roots, you should do it as
.. x^2 + bx^2 + cx + d
.. = ((x + b)x + c)x + d ... "Horner's form" of the polynomial
Evaluation in this form is similar to synthetic division.

For the first one, using x=4, you get
.. ((4 -3)4 - 24)4 + 42 = (4-24)4 + 42 = -20*4+42 (obviously not zero)