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^4+5x^3-2x^2+5x+6; x+3

Accepted Answer

Welcome back. I am so glad you're here. We're asked for the following polynomial to perform factor theorem and synthetic division to identify whether X plus nine is a factor or not. Our polynomial is F of X equals six, X to the fourth plus 35 X cubed minus 317 X squared minus 1419 X minus 945. In our answer, choices are answer choice. A yes, in answer. Choice. B no. All right. This one needs a calculator. So how do we do this? The important part is knowing how to set it up. So if we're trying to figure out if X plus nine is a factor of this to use the factor theorem in synthetic division, what we need to know is that we are going to be tech testing for X minus A and if X plus nine is X minus A, our A is going to be negative nine and we're going to use negative nine to test for synthetic division. And when we set this up, we're going to take all of the coefficients in the constant term from our polynomial. So we'll have six 35 negative 317 negative 1419 and negative 945. We're going to use the synthetic division that we recall from previous lessons and test this out. So we start off bringing down this first one, bringing down the six. And then we're going to multiply the negative nine multiplied by six which is a negative 54. And then we're going to add 35 plus negative which is negative 19. Then we multiply negative nine multiplied by negative 19 which is a positive 171. Then we add negative 317 plus 171 is a negative 146. Then we multiply negative nine multiplied by negative 146 is a positive 1314. Then we add negative 1419 plus 1314 is a negative 105. Then we multiply negative nine multiplied by negative 105 is a positive 945. And then we add negative 945 plus 945 is zero. And so our remainder theorem tells us that when we have a remainder of zero, that this is a factor this X plus nine. So negative nine is a zero and X plus nine is a factor of this polynomial. So we look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

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Use the factor theorem and synthetic division to determine whether the second polynomial is a factor of the first. See Example 1. 2x^4+5x^3-2x^2+5x+6; x+3