Suppose you were asked to solve the following two problems...

Suppose you were asked to solve the following two problems on a test. (a) Find the remainder when 9x1000 − 14x562 + 16x + 22 is divided by x + 1. (b) Is x − 1 a factor of x567 − 7x400 + x9 + 6? Obviously, it's impossible to solve these problems by dividing, because the polynomials are of such large degree. Use one or more of the theorems in this section to solve these problems without actually dividing. (a) Find the remainder when 9x1000 − 14x562 + 16x + 22 is divided by x + 1. (b) Is x − 1 a factor of x567 − 7x400 + x9 + 6? YesNo

Transcript

00:01 So in this problem, we're being asked to use some of the theorems that you learn during this chapter.

00:04 So in part a, we're being asked to find the remainder when the given polynomial function is getting divided by x plus 1.

00:11 Well, because we're just looking for the remainder, we can use the remainder theorem, which says that if we divide the polynomial f of x, and we divide it by the binomial x minus c, then the remainder is going to be f of c.

00:27 So in this case, our c value would equal to negative 1.

00:31 So to find our remainder, we just need to find f of negative 1.

00:34 So we'll substitute negative 1 in place of x.

00:37 So we'll have 4 times negative 1 raised to the 1 ,000 power, minus 18 times negative 1 raised to the 560 second power, plus 12 times negative 1 plus 28.

00:49 Well, let's start with our exponents.

00:51 Well, negative 1 raised to the 1 power.

00:53 Well, 1 raised to any power is 1, and the negative is getting raised to an even power.

00:57 Which means it's going to be positive 1.

01:00 So then we'll have 4 times 1, which is just 4.

01:03 Next, for negative 1 raises to 560 second power.

01:07 Again, the hour is even, so therefore it's going to be positive, and 1 to any power is 1.

01:13 So negative 18 times 1 is still going to be negative 18.

01:16 Then we have 12 times negative 1, that's negative 12, and then we'll bring down the 18.

01:20 Well, 4 minus 18 is negative 14.

01:24 Then we'll have negative 14 minus 12, which is negative 26, and negative 26 plus 8 is negative 18.

01:31 So now we've found that the remainder will be negative 18...

Question

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