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 x³?

The answer is x² [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):

………… x²

(x – 1) |(x³ – 2x + 1)

Now, we have to multiply x² by both x and – 1 like so (in teal):

……….. x²

(x – 1) |(x³ – 2x + 1)

……….. x³ – x²

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

……….. x²

(x – 1) |(x³ – 2x + 1)

………. -(x³ – x²)

…………….. x² – 2x + 1

Now, we ask ourselves: how many times does x go into x²? 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.