We can use synthetic division when dividing or evaluating polynomials. To evaluate polynomials using synthetic division, we use the same process as dividing polynomials with synthetic division. When evaluating a polynomial using synthetic division, the remainder is the answer that you would arrive at if you evaluated by plugging in.
Using synthetic division to evaluate a polynomial. So, behind me I have a fourth degree polynomial. Okay, and we're asked to find what p of -4 is, okay? So before what we could have done is plug in -4. So this is going to be equal -4 to the fourth minus 5, negative fourth squared plus 4 times -4 plus 12, okay? That's doable but we're going to get to some pretty big numbers. -4 to the fourth is not an easy number to remember. So there is another way we can do this and that's using synthetic division. So what we do is we take the number that we're trying to evaluate, -4 in this case that's going to go outside of our synthetic division bracket. Draw your bracket and continue with the synthetic division process. So right are the coefficients of our functions in descending order making sure that all of our number, degrees are taken care of. so this particular case we don't have an x cube term, so we need to put in a zero. Zero, -5, 4 and 12. Going through the synthetic division process. Drop down your 1, multiply and add throughout the entire thing. So -4 times 1 is -4 and add -4 -4, -4 times -4 is 16, -5 plus 16 is 11, -4 times 11 is -44, 4 plus -44 is -40, -4 times -40 is 160, finally ending up with 172. Okay. So we put our number on the outside and then [IB] division. In order to evaluate this function. the only number we're actually concerned with is that remainder. Okay, so what we actually did is this entire process just to get that last number, 172. In this case p of -4 is going to be 172. If we calculate this all out, this would in fact turn into 172, okay? So synthetic division to evaluate a polynomial function can actually reduce your workload. Instead of having to deal with powers of big numbers we can turn it into simple multiplication and addition to get our answer.