Question

Write a linear factorization of the function. f(x) = x4 + 4x2

Answer 1

Can you factor a common factor out?

Answer 2

x

Answer 3

Yes, but you can factor more than that.

Answer 4

x^2?

Answer 5

Correct. Start by factoring out x^2; what do you get?

Answer 6

x^2(x^2-4) right?

Answer 7

Wasn't the original polynomial f(x) = x4 + 4x2?
(with a + between the terms?)

Answer 8

oh yes i meant +

Answer 9

We're here, then.
f(x) = x^2(x^2 + 4)

Answer 10

yes what do i do from there

Answer 11

We can write the first two terms as linear terms:
\(f(x) = x \cdot x \cdot (x^2 + 4) \)

Answer 12

Now we need to deal with \(x^2 + 4\) and make it the product of two linear terms.

Answer 13

If we had \(x^2 - 9\), can you factor this?

Answer 14

(x+3)(x-3)

Answer 15

Correct.
Now let's write \(x^2 + 4\) as a difference of squares. Then you'll be able to do the same kind of factoring.


Notice that \(x^2 - 9 = (x + \sqrt 9) (x - \sqrt 9) = (x + 3)(x - 3) \)

Now let's do the same to this.
Use the square root as I did above in the middle step.
\(x^2 - (-4)\)

Answer 16

so (x+squareroot of 2)(x-squareroot of 2)

Answer 17

You're on the right track, but you need to be careful.
Let's follow the example carefully.

\(x^2 - (-4) = (x + \sqrt{-4} )(x - \sqrt{-4} )\)

Now we deal with the imaginary roots.

\(= (x + i\sqrt{4} )(x - i\sqrt{4} )\)

Now we deal with \(\sqrt 4\)

\(= (x + 2 i )(x - 2i)\)

Answer 18

so would this be the answer f(x) = x^2 (x + 2i)(x - 2i)

Answer 19

Yes. Unless you want to split up the x's and show it as

\(f(x) = x \cdot x \cdot (x + 2i)(x - 2i) \)

Now you are really showing every linear term.

Answer 20

Ok thank you! i understand now.

Answer 21

Great.
You're welcome.