Question

How do you factor 16x^4-81?

Answer 1

Notice it has only two terms.
Which types of two-term polynomials do you know how to factor?

Answer 2

I dont really remember. We have only reviewed trinomials.

Answer 3

Ok.
The difference of two squares is a two-term polynomial that can be factored.
Also, the sum and difference of two cubes can also be factored.

Factorization of the difference of two squares:
\(a^2 - b^2 = (a + b)(a - b)\)

Factorization of the sum of two cubes:
\(a^3 + b^3 = (a + b)(a^2 - ab + b^2)\)

Factorization of the difference of two cubes:
\(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)

Now look at your problem. Which of the above does it resemble?

Answer 4

The square root of 16=4, x^4=x^2, and 81=9

Answer 5

Great.
Since \(x ^4 = (x^2)^2\), and \(81 = 9^2\), this certainly looks like a difference of squares.
Now use the difference of squares formula above to factor it.
What do you get?

Answer 6

(4x^2-9)(4x^2+9)?

Answer 7

yep. that's the right answer.

Answer 8

Great. So far you are correct, but you are not done yet.

Answer 9

Cant it go farther? like 2 and 3?

Answer 10

Factoring means factoring completely.
Now you need to look at each factor again, and see if it's factorable.

Answer 11

(2x-3)(2x+3)

Answer 12

Can \(4x^2 - 9\) be factored?
Can \(4x^2 + 9\) be factored?
Once again, you are dealing with only two terms, so you need to look at the possibility of difference of two squares and sum or difference of two cubes.

Answer 13

Wow, you're ahead of me.
Great job.
\(4x^2 - 9\) does factor like you did, but \(4x^2 + 9\) is a sum of squares which is not factorable.

Answer 14

Now you can write the complete factorization.

Answer 15

So the final answer is (2x-3)(4x^2+9)

Answer 16

No.
Let's look at it one step at a time, but keeping everything as we go along.
I'll color code it so you can see how this problem should be done.

Answer 17

Question:
Factor \(16x^4-81\).

Solution:

\(\color{red}{16x^4-81}\)

\(= \color{red}{(4x^2 - 9)(4x^2 + 9)} \)

I rewrite the previous step to show what will be factored in green.

\(= \color{green}{(4x^2 - 9)}(4x^2 + 9) \)

\(= \color{green}{(2x - 3)(2x + 3)}(4x^2 + 9) \)

Final answer: \( (2x - 3)(2x + 3)(4x^2 + 9) \)

Answer 18

Oh. @mathstudent55 that makes sense