Math  /  Algebra

Question14) x416x^{4}-16

Studdy Solution

STEP 1

What is this asking? We need to factor the expression x416x^4 - 16 completely. Watch out! Don't stop factoring too early!
Remember to look for **difference of squares** patterns.

STEP 2

1. Factor as a difference of squares.
2. Factor the remaining difference of squares.

STEP 3

Hey everyone!
We've got x416x^4 - 16, and guess what?
It's a **difference of squares**!
Remember, a difference of squares looks like a2b2a^2 - b^2, and it factors into (a+b)(ab)(a+b)(a-b).

STEP 4

In our case, a=x2a = x^2 and b=4b = 4 because (x2)2=x4(x^2)^2 = x^4 and 42=164^2 = 16.
So, we can rewrite our expression as (x2)242(x^2)^2 - 4^2.
Using the difference of squares formula, we get (x2+4)(x24)(x^2 + 4)(x^2 - 4).
Awesome!

STEP 5

Look closely!
We have another **difference of squares** hiding in our result: x24x^2 - 4.
This time, a=xa = x and b=2b = 2.

STEP 6

Applying the difference of squares formula to x24x^2 - 4, we get (x+2)(x2)(x+2)(x-2).
The x2+4x^2 + 4 term doesn't factor any further using real numbers.

STEP 7

Putting it all together, our completely factored expression is (x2+4)(x+2)(x2)(x^2 + 4)(x+2)(x-2).
Boom!

STEP 8

The fully factored form of x416x^4 - 16 is (x2+4)(x+2)(x2)(x^2 + 4)(x+2)(x-2).

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