Math  /  Algebra

QuestionFactor completely. 364z236-4 z^{2}

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression as a product of simpler expressions. Watch out! Don't forget to factor *completely*!
Sometimes, there are multiple steps involved.

STEP 2

1. Factor out the greatest common factor.
2. Use the difference of squares.

STEP 3

Let's **look** at the expression 364z236 - 4z^2.
Both terms are divisible by **4**, so we can **factor out** a 4!

STEP 4

364z2=494z2=4(9z2)36 - 4z^2 = 4 \cdot 9 - 4 \cdot z^2 = 4(9 - z^2) Remember, factoring out a 44 is the same as dividing each term by 44.
We can always check our work by distributing the 44 back in!

STEP 5

Now, we have 4(9z2)4(9 - z^2).
Inside the parentheses, we have a **difference of squares**!
Remember, a2b2=(a+b)(ab)a^2 - b^2 = (a+b)(a-b).
Think of 99 as 323^2 and z2z^2 is already in perfect square form.

STEP 6

So, we can rewrite 9z29 - z^2 as 32z23^2 - z^2, which factors to (3+z)(3z)(3+z)(3-z).

STEP 7

Putting it all together, we get: 4(9z2)=4(3+z)(3z)4(9 - z^2) = 4(3+z)(3-z) We can also write this as 4(3z)(3+z)4(3-z)(3+z), since multiplication is commutative!

STEP 8

The completely factored form of 364z236 - 4z^2 is 4(3z)(3+z)4(3-z)(3+z).

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