Question 1206506: use synthetic division to determine the answer.
6. Which one of the binomials below is a factor of the polynomial x^3 + 4x^2 + x - 6?
x + 4
x - 4
x + 3
x - 3
x - 2
x + 1
Found 2 solutions by josgarithmetic, math_tutor2020:
Answer by josgarithmetic(39620)   (Show Source):
You can put this solution on YOUR website! A matter of performing the synthetic division to test for roots 1, 2, 3, 6, -1, -2, -3, -6. But each time you find one, you have only one less to look for.
Answer by math_tutor2020(3817) (Show Source):
You can put this solution on YOUR website!
Answer: x+3 only
Explanation
Let p(x) = x^3 + 4x^2 + x - 6
This is what the synthetic division table looks when dividing p(x) over (x+4).
x = -4 is the test root.
Ignoring the -4 in the top left corner, the other values in the top row are the coefficients of p(x).
I won't explain the process of how synthetic division works (that's for other internet resources to do).
This page is simply to help you check your work to make sure you're on the right track.
The last value in the bottom row is the remainder.
The nonzero remainder means x+4 is NOT a factor of x^3 + 4x^2 + x - 6
This rules out choice A.
Nonzero remainders also happen for choices B, D, E, and F. They are eliminated as well. I'll leave the scratch work for the student to do.
On the other hand, x+3 is a factor of x^3 + 4x^2 + x - 6 because we get a zero remainder.
Side notes:- If you did not have a list of multiple choice answers, then you would have to use the rational root theorem to generate the list of all possible rational roots, which in turn could be used in synthetic division.
- We can verify x+3 is a factor of x^3 + 4x^2 + x - 6 by use of a graphing calculator. One of the x intercepts is x = -3.
- x^3 + 4x^2 + x - 6 fully factors to (x-1)(x+2)(x+3)
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