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

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+6x^2-2x-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 our 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 a yes or answer B no. So our first step here is to recall the factor theorem which states that if X minus A is a factor of the function F of X, then we know that F of A is equal to zero. So here we are asked to verify that the expression X plus two is a fact of F of X. And so from X plus two, we see that our value for A is negative two. And so we are really trying to verify that F of negative two is equal to zero. So now to find our value of F of negative two and thus verify this statement, we need to use synthetic division. So setting up for synthetic division, we first have a vertical line and then a horizontal line which passes through it. And so on the left of the vertical line, we place the value of A which is negative two. And then to the right, the vertical line, we place each of our coefficients from our given function starting with the highest power of X here, the highest power is three. So the coefficient of X cubed is our first coefficient which is just one. Now moving on to the coefficient of X squared, we have four, moving on to X have negative 11. And lastly the coefficient of X to the zeroth power gives us negative 30. So now with our tables set up, we can begin performing synthetic division. And so the first step is to just bring down our first coefficient which is one and rewriting it underneath the horizontal line, we then take this value of one and to multiply it by the value of A which is negative 21 times negative two gives us negative two. And now we take this product and add it to the coefficient in the second column. Four plus negative two gives us two. We then repeat this process now taking two and multiplying it by negative two, this gives us negative four which we add to the coefficient in the third column. So negative 11 plus negative four gives us negative 15, repeating this process one more time. We now take negative 15 times negative two which gives us positive 30. We then add this product to the last coefficient. So we have negative 30 plus 30 which gives us zero. And now our value for F of negative two is given by the last value from synthetic division. So here we see that the value of zero is F of negative two. So we have confirmed that F of negative two is in fact equal to zero. So according to the factor theorem, we have verified that X plus two is in fact a factor of our function F of X. Therefore, we are left with answer choice. A where yes, X plus two is a factor of our given 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+6x^2-2x-7; x+1

Key Concepts

Factor Theorem

Synthetic Division

Polynomial Functions