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^3-5x^2+3x+1; x-1

Accepted Answer

Hey, everyone here we are asked to perform factor theorem and synthetic division to identify whether X minus three is a factor or not of the given function F of X is equal to X cubed plus two, X squared minus five, X minus six. Here we have two answer choice options, answer choice A yes and answer B no. So here our first step is to recall the factor theorem which states 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 here we are asked to verify the expression X minus three. And so from X minus three, we see that our value for A is three. And so we are actually trying to verify that F of three is in fact equal to zero. So to verify this statement, we need to perform synthetic division. And so to start setting up synthetic division, we first draw a vertical line and then a horizon line that passes through it. And to the left of the vertical line, we place the value for A which is three and then to the right of the vertical line, we place each of the coefficients given by our function starting with the highest power of X here, the highest power is three. So we start with the coefficient of X cubed which is one moving on to X squared, we have the coefficient of two moving on to X, we have a coefficient of negative five. And then lastly, the coefficient of X to the zeroth power is negative six. So now with our table set up for synthetic division, we can begin the process by first bringing down the first coefficient which is one and rewriting it underneath the horizontal line, we then take one and multiply it by the value of A which is 31 times three gives us three. And now we take this product and add it to the coefficient in the second column. So we have two plus three which gives us five. Then we take this value of five and repeat the process by multiplying it by three. This gives us 15. And then we add this product to the coefficient in the third column. So negative five plus 15 gives us 10, repeating this process one last time. We now take 10 and multiply it by three. This gives us 30 of which we add to our last coefficient. So negative is six plus 30 gives us 24. And now this last value of 24 that we obtained from synthetic division tells us our value of F of three. So F of three is equal to 24. And this tells us that F of three is clearly not equal to zero. Therefore, according to the factor theorem, we know that X minus three is not a factor of our given function. So since again, F of three is not equal to zero, we are left with answer choice B where the answer is no, X minus three is not 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^3-5x^2+3x+1; x-1

Key Concepts

Factor Theorem

Synthetic Division

Polynomial Functions