Question

How do you factor `35x ^ { 5} - 21x ^ { 3} + 28x`?

Answer 1

The factored form is `7x(5x^4-3x^2+4)`

Explanation:

First, take out any factors which all three terms share.

`35x^5 - 21x^3 + 28x`

We can pull out a factor of `color(blue)(7x` from all terms.

`color(white)"XXX"35x^5 color(white)".."- color(white)"..."21x^3 color(white)".."+ color(white)".."28x`
`= color(blue)(7x) * 5x^4-color(blue)(7x) * 3x^2+color(blue)(7x) * 4`
`= color(blue)(7x) * (5x^4-3x^2+4)`

Now, to factor `5x^4-3x^2+4`, we need to treat `color(red)(x^2)` like `x` in a quadratic, and factor it like a regular quadratic.

`color(white)"XXX"5x^4-3x^2+4`
`= 5(color(red)(x^2))^2 - 3(color(red)(x^2))+4`

To factor quadratics where `a` is not 1, first multiply `a` by `c`, and find two factors of `ac` which add up to `b`. However, this is as far as we need to go for this problem, since there are no factors of `20` which could possibly add up to `-3`.

This means that `5x^2-3x+4` is not factorable. Therefore, `5(color(red)(x^2))^2 - 3(color(red)(x^2))+4` is also not factorable.

Therefore, the factored form of `35x^5 - 21x^3 + 28x` is `7x(5x^4-3x^2+4)`.