How do you factor out x^4 - 16?

Question

yea

Answer 1

You asked "how" so I'm going explain the philosophy...

You probably already know how to use FOIL to multiply to sets of polynomials. FOIL stands for First, Outer, Inner, Last. Factoring is just the opposite.

So you can start by writing the parentheses.

(.........)(........)

Look at the first term. What two terms can you multiply together to get x^4. The most obvious one is x^2 * x^2. So add these to your parentheses.

(x^2.....)(x^2....)

Now since the last number is negative then you need one of each sign in either set. If it was positive you would need either 2 "+" signs or 2 "-" signs

(x^2+...)(x^2-..)

The last term is 16. What can you multiply together to get 16? 1*16, or 2*8, or 4*4.

The other thing you need to think about is the sum required to get to the middle term. Since there is no middle term in this example, the sum is zero.

If you used 2 & 8 as the digits you would get 2x^2-8x2 = -6x^2 or 6x^2 depending on the order. If you used 1 & 16, would get x^2-16x^2=-15x^2 or 15x^2 if you reversed the order. What happens if you use 4? You get 4x^2-4x^2 = 0, the term we were looking for. So the answer is:

(x^2+4)(x^2-4)

Answer 2

x^4 - 16 = (x² - 4)(x² + 4) = (x - 2)(x + 2)(x² + 4)

Answer 3

You could look at it as this

(x^2 + 4) * (x^2 - 4) = x^4 - 16

(x^2 + 4) * ((x + 2) * (x - 2)) = x^4 - 16

Answer 4

x^4 - 16
=(x^2-4)(x^+4)
= (x-2)(x+2)(x^2+4)
Many peple forgot to also factor the (x^2-4). You must do this to get a complete factoring.

Answer 5

this would be an example of difference of perfect squares since x^4 is (x^2)^2 and 16 is 4^2 . the form for diff of perf sq is :
x^2 - y^2 = (x-y)(x+y)

so here x=x^2 and y=4
(x^2 - 4) (x^2 + 4)

now...this is not done because (x^2 - 4) is also diff of perf sq...so it would factor again to (x-2)(x+2).

your answer would then be:
(x-2)(x+2)(x^2 + 4)

Answer 6

x^4 - 16 = (x^2 + 4 )(x^2 - 4)

= ( x^2 + 4 ) ( x -2 )( x +2 )

Note: Use special product formula.... Difference of two squares
x^2 - y^2 = (x +y )(x - y).

Answer 7

First: express both terms in lowest terms....

(x^2)(x^2) - (4)(4)

Sec: you have the difference of squares --- combine one term from the 1st set of parenthesis with the other term from the 2nd set of parenthesis...

(x^2 + 4)(x^2 - 4)

Third: factor the 2nd set of parenthesis --- you have the difference of squares...

= (x^2 + 4)(x + 2)(x - 2)

Answer 8

(x^4-16) = (x^2 + 4) * (x^2 - 4)

Answer 9

(x^4 - 16)
= (x^2 - 4) (x^2+4)
= x^4 + 4x^2 - 4x^2 - 16
= x^4 - 16 is the Answer

Answer 10

oooh, look at this, a square differance... x^4-16 = (x^2)^2 - 4^2 = ..... guess what, yes you got it right: x^2 - 4 multiplied by what?: x^2 + 4 !!!!

(x^2-4)(x^2+4) = x^4-16 always!