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.
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!
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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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