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. 2x^3+x+2; x+1

Accepted Answer

Hey, everyone here we are asked to perform factor theorem and synthetic division to identify whether X plus three is a factor or not of the given polynomial function F of X is equal to nine X cubed plus 11 X plus 50. Here we have two answer, choice options, answer choice A yes or answer B no. So here to begin, we need 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 identify that X plus three is a factor of our given function. And so from X plus three, we get that A is negative three. So we need to verify that F of negative three is equal to zero. Now, to verify this statement here, we need to perform at synthetic division. So setting up for a synthetic division, we first have a vertical line and then a horn that passes through it. And to the left of the vertical line, we place the value of A which is negative three and then to the right of the vertical line, we need to place each of the coefficients from our function F of X starting with the highest power of X. Well, the highest power is three. So the coefficient of X cubed is nine and it is the first coefficient that we write on our table. Now we move on to the next highest coefficient which is X squared. What we see X squared does not appear in our function. So its coefficient is just zero. Then moving on to X, we see X has a coefficient of 11 and lastly moving on to X to the zeroth power or just our constant value here, our last coefficient is just 50. So now that we have it finished setting up, we can go ahead and begin performing synthetic division. And so the first step is to just bring down the first coefficient which is nine and rewrite it underneath the horizontal line. Now we take this value of nine and to multiply it by the value of a which is negative three. Now simplifying we get negative 27 now we take this product and add it to the coefficient in the second column which is zero. So zero plus negative 27 gives us negative 27. We now repeat this process but instead take negative multiply it by negative three. Simplifying, we get positive 81 we now take this product and add it to the coefficient in the third column. So we have 11 plus 81 giving us repeating this process one more time. We have 92 times negative three, which simplifies to negative 276 which we add to the last coefficient which is 50 50 plus negative 276 gives us negative 226. And now this last value negative 226 is our F of negative three value. And will we see that this value is not zero? So we confirm that F of negative three does not equal zero. And this tells us that our given factor S X plus three is not a factor of our given function F of X because again, F of negative three does not equal zero. So again, we are left with the answer being no, X plus three is not a factor. So we are left with answer choice B where again, X plus three is not a factor of our given polynomial 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. 2x^3+x+2; x+1

Key Concepts

Factor Theorem

Synthetic Division

Polynomial Remainder Theorem