SOLUTION: Please help me factor this equation: {{{x^(2n+1)-2x^(n+1)+x}}} Thank you.

Question 206554This question is from textbook Introductory Algebra
: Please help me factor this equation:
Thank you.
This question is from textbook Introductory Algebra

Found 3 solutions by Alan3354, jim_thompson5910, stanbon:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!

=
=
Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
The last solution is correct, but here's why factors to ...

Start with the given expression.

Rewrite as . Note:

Now let (a substitution to make things easier)

Replace each with 'y'

Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .

Now multiply the first coefficient by the last term to get .

Now the question is: what two whole numbers multiply to (the previous product) and add to the second coefficient ?

To find these two numbers, we need to list all of the factors of (the previous product).

Factors of :
1
-1

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .
1*1 = 1
(-1)*(-1) = 1

Now let's add up each pair of factors to see if one pair adds to the middle coefficient :

First Number Second Number Sum
1 1 1+1=2
-1 -1 -1+(-1)=-2

From the table, we can see that the two numbers and add to (the middle coefficient).

So the two numbers and both multiply to and add to

Now replace the middle term with . Remember, and add to . So this shows us that .

Replace the second term with .

Group the terms into two pairs.

Factor out the GCF from the first group.

Factor out from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.

Combine like terms. Or factor out the common term

Condense the terms.

Plug in

So factors to .

In other words, .

Answer by stanbon(75887)   (Show Source): You can put this solution on YOUR website!
x^(2n+1)-2x^(n+1)+x
Factor out "x" to get:
x[(x^n)^2 - 2 x^n +1]
---
Let x^n = k
---
Substitute to get:
x[k^2 -2k + 1]
---
= x(k-1)^2
---
Substitute to get:
---
= x((x^n)-1)^2
---
Cheers,
Stan H.
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