Use the factor theorem and synthetic division to determine whe...

Question

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.

$x^{3} + 6 x^{2} - 2 x - 7; x + 1$

Accepted Answer

Hey, everyone here we are asked to perform factor theorem and synthetic division to identify whether X plus two is a factor or not of the given function F of X is equal to X cubed plus four, X squared minus 11 X minus 30. Here we have two answer, choice options, answer choice A yes or answer B no. To begin this problem, we first want to recall the factor theorem which tells us that if X minus A is in fact a factor of the function F of X, then we know that F of A is equal to zero. So in our case, we know that if X plus two is a factor of our function F of X, then we know that F of negative two is going to be equal to zero. So now we want to utilize synthetic division to verify that F of negative two is equal to zero. So setting up our table for synthetic division to the left of the vertical line, we place the value for A which is negative two. And then to the right of the vertical line, we list all of our coefficients starting with X cubed which has a coefficient of one. Then our next coefficient is four, then we have negative 11. And lastly, we have negative 30. So now beginning to perform synthetic division, the first step is to bring down the first coefficient which is one and rewrite it underneath the horizontal line. Then we bring up this value and we multiply it by the value of A which is negative two. Now the product of one and negative two gives us negative two of which we add to the coefficient in the second column. So we have four plus negative two which gives us positive two. Now repeating this process, we take two and we multiply it by negative two, this gives us negative four. Then we have negative 11 plus negative four which gives us negative 15. Now repeating the process one more time, we take negative 15 and multiply it by negative two, this gives us positive 30. We now have negative 30 plus positive 30 which cancels out and gives us zero. So now our value for F of negative two is the value of our remainder. And we know that when we perform synthetic division that our remainder is the last value. So here we see that F of negative two is going to be zero. And now, since we have found that F of negative two is in fact equal to zero, we know that X plus two is in fact a factor of our given function So we are left with answer choice. A where we now know that yes, X plus two is a factor of our function. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.
$x^{3} + 6 x^{2} - 2 x - 7; x + 1$

Key Concepts

Factor Theorem

Synthetic Division

Polynomial Factorization

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.x3+6x2−2x−7;x+1x^3+6x^2-2x-7; x+1

Key Concepts

Factor Theorem

Synthetic Division

Polynomial Factorization

Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1.
$x^{3} + 6 x^{2} - 2 x - 7; x + 1$