how would you simplify y= (x^5-32)/(x-2)?

I know that it would equal (x^5-2^5)/(x-2), but what's the next step?

It wouldn't be just (x^4-2^4), would it?

kristie, in general I think you can prove (x-a)|(x^n - a^n); read (x-a) divides the expression on the right of '|'. You should be able to prove easily (x-a)*(x^(n-1) +ax^(n-2) +a^2 x^(n-3) + ... + a^(n-1)) = (x^n - a^n).
Here we have (x-2)|(x^5 - 2^5). Can you supply the details?
(Hopefully I recalled the correct identity; check this, it should be in your text.)

y= (x^5-32)/(x-2)?
You need to factor the numberator. which is likely not a easy task. However, if one of the factors is 2, you can make some headway. Checking to see if two is a factor..
y= (2^5-2^5)/(2-2)= ? 0/0 so it is not certain yet if y=0, and x=2 is a factor.
So divide the numberator by the denominator...
I did synthetic division to see if x=2 is a do the division the way you learned..

y= (x-2)(polynomial)/(x-2) and the x-2 factors divid out.

kristie you need to check the division I did, it is easy to make errors on those.

I am still confused about what I need to do...

Divide x-2 into the numerator, see if it goes evenly.

OK. I'll try that. Thank you so much for your help!

Explain Bot Explain Bot answered
9 months ago

You're welcome! To continue simplifying the expression y = (x^5-32)/(x-2), you can use polynomial long division or synthetic division to divide the numerator (x^5 - 32) by the denominator (x - 2).

Here's how you can do it using polynomial long division:

Step 1: Start by dividing the first term of the numerator, which is x^5, by the first term of the denominator, which is x. This gives you x^4.

Step 2: Multiply x^4 by the denominator (x - 2), and subtract the result from the numerator (x^5 - 32). This will give you a new polynomial.

Step 3: Repeat steps 1 and 2 with the new polynomial until you've divided all the terms of the numerator.

Step 4: The result of the division will be the quotient, and any remaining terms will be the remainder.

In this case, when you divide x^5 - 32 by x - 2, you will find that the quotient is x^4 + 2x^3 + 4x^2 + 8x + 16, and there is no remainder.

So, the simplified form of y = (x^5 - 32)/(x - 2) is y = x^4 + 2x^3 + 4x^2 + 8x + 16.

I hope that helps! Let me know if you have any further questions.