Solving Polynomial Remainders

[Ambushes]

Homework Statement

Find each remainder:

a. (x^3 + 5x^2 - 7x + 1) ÷ (x+2)(x-1)b. (2x^3 + x^2 - 4x - 2) ÷ (x^2 + 4x + 3)

Homework Equations

N/A. (We've used Long Division and Synthetic Division for previous questions.)

The Attempt at a Solution

How would i go about solving these? I'm pretty stuck.

 

[tiny-tim]

Hi Ambushes!

Why can't you use long division?​

 

[Ambushes]

Hi Ambushes!

Why can't you use long division?​

I've never done long division with two binomials/zero's before, usually it's only one like (x+2), etc.
What do i have to do differently?

Thanks for the help!

 

[tiny-tim]

… usually it's only one like (x+2), etc.
What do i have to do differently?

nothing!

it's only like the difference between "ordinary" long division by 2 and by 23

 

[Ambushes]

nothing!

it's only like the difference between "ordinary" long division by 2 and by 23

Haha, wasn't as hard as i thought. Got the right answer =).
If you don't mind, i have one more question. It's probably really easy, but how do i know when to use Synthetic Division, and when to use Long Division?

 

[tiny-tim]

… how do i know when to use Synthetic Division, and when to use Long Division?

hmm … I've never come across synthetic division before

i had a quick look at it on wikipedia, and my impression is that long division would always be easier, and less likely to lead to mistakes

does anyone who's actually used it want to chip in? ​

 

[rcgldr]

For 1 a, why not multiply the two divisor terms to produce a product of a single polynomial (x^2 + ...) and then continue using long division?

 

[Dick]

hmm … I've never come across synthetic division before

i had a quick look at it on wikipedia, and my impression is that long division would always be easier, and less likely to lead to mistakes

does anyone who's actually used it want to chip in? ​

Synthetic division is just a streamlined form of the long division algorithm. It does exactly they same thing as long division but omits writing a lot of powers of x. It's not in any way a different thing. There's no need to decide which to use. Use synthetic division if you remember how to do it. Otherwise long division will give you the same answer.

 

[tiny-tim]

thanks, Dick!