Math Snap

STEP 1

What is this asking?
We need to simplify the expression $392 x^{4}-128 x^{2} y^{2}$ by factoring out as much as possible, and if it can't be factored, we'll say it's prime!
Watch out!
Don't rush!
It's easy to miss a factor or two if we don't carefully consider all possibilities.

STEP 2

1. Find the greatest common factor of the coefficients.
2. Find the greatest common factor of the variables.
3. Factor the difference of squares.

STEP 3

Let's find the greatest common factor (GCF) of the coefficients, which are 392 and 128.
We can do this by listing the prime factors of each number.

STEP 4

$$392 = 2 \cdot 196 = 2 \cdot 2 \cdot 98 = 2 \cdot 2 \cdot 2 \cdot 49 = 2 \cdot 2 \cdot 2 \cdot 7 \cdot 7 = 2^3 \cdot 7^2$$ $$128 = 2 \cdot 64 = 2 \cdot 2 \cdot 32 = 2 \cdot 2 \cdot 2 \cdot 16 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 8 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 4 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 2^7$$

STEP 5

The common prime factors are three twos, so the GCF is $2^3 = 8$.
Awesome!

STEP 6

Now, let's look at the variables.
We have $x^4$ and $x^2 y^2$.
The common factors are $x^2$.
So, the GCF of the variables is $x^2$.

STEP 7

We've found that the GCF of the whole expression is $8x^2$.
Let's factor that out:
$$392 x^{4}-128 x^{2} y^{2} = 8x^2(49x^2 - 16y^2)$$ Look closely at what's left inside the parentheses!

STEP 8

We have $49x^2 - 16y^2$, which is a difference of squares because $49x^2 = (7x)^2$ and $16y^2 = (4y)^2$.
Remember the difference of squares formula: $a^2 - b^2 = (a+b)(a-b)$.

STEP 9

Applying this formula, we get:
$$49x^2 - 16y^2 = (7x)^2 - (4y)^2 = (7x + 4y)(7x - 4y)$$

STEP 10

Putting it all together, our completely factored expression is:
$$392 x^{4}-128 x^{2} y^{2} = 8x^2(7x + 4y)(7x - 4y)$$

A. $392 x^{4}-128 x^{2} y^{2}= 8x^2(7x + 4y)(7x - 4y)$