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Polynomial Long Division

Q:  How do I determine the quotient and the remainder, r(x), in the division of a(x) by b(x) when:

a(x) = x³ – 2x +1

and

b(x) = x-1  ?


A:  Ok, we are dividing (x – 1) into (x³ – 2x + 1)

Step 1:  We ask start our with the first term and ask ourselves: how many times does x go into ?

The answer is [this is the first part of our answer, or the quotient]….

I am attempting to “format” this to look like a division problem (the answer, in purple, will just be above the division problem):

…………

(x – 1) |(x³ – 2x + 1)

Now, we have to multiply by both x and – 1 like so (in teal):

………..

(x – 1) |(x³ – 2x + 1)

……….. x³ – x²

Now, we subtract the teal from the red and bring down extra terms:

………..

(x – 1) |(x³ – 2x + 1)

………. -(x³ – x²)

…………….. x² – 2x + 1

Now, we ask ourselves: how many times does x go into ? The answer is x. Put that with our answer, and as before, multiply it with x and -1 like so:

……….. x² + x

(x – 1) |(x³ – 2x + 1)

………. -(x³ – x²)

…………….. x² – 2x + 1

……………. x² – x

Now, subtract teal from red:

……….. x² + x

(x – 1) |(x³ – 2x + 1)

………. -(x³ – x²)

…………….. x² – 2x + 1

…………….-(x² – x)

…………………….-x + 1

And again, divide x into -x, which is -1, multiply it down (the teal).. then subtract it:

…………x² + x – 1

(x – 1) |(x³ – 2x + 1)

………. -(x³ – x²)

…………….. x² – 2x + 1

…………….-(-x² – x)

…………………….-x + 1

……………………..-(-x + 1)

……………………………….0

Ok!

So, the quotient is: x² + x – 1 and the remainder is .0.

2 thoughts on “Polynomial Long Division

  1. Posted this in a hurry, so didn’t get to mess with better formatting or check for minor errors yet! Will when I get home!

  2. FYI: Due to rushing (because of a screaming baby), there was an algebra error in this initial post — which affected all the numbers! The error has been fixed and it is all good now.

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