Question Factor the expression $z^{4} - 16$ . If the expression is prime, enter PRIME.

Studdy Solution

STEP 1

Assumptions
1. We are given the expression $z^{4} - 16$ .
2. We need to factor the expression completely.

STEP 2

Recognize that the expression is a difference of squares. A difference of squares can be factored using the formula $a^2 - b^2 = (a + b)(a - b)$ .

STEP 3

Identify $a^2$ and $b^2$ in the given expression. In this case, $a^2 = z^4$ and $b^2 = 16$ .

STEP 4

Find $a$ and $b$ by taking the square root of $a^2$ and $b^2$ .
$a = \sqrt{z^4} = z^2$ $b = \sqrt{16} = 4$

STEP 5

Apply the difference of squares formula to the expression $z^{4} - 16$ using the values of $a$ and $b$ found in Step 4.
$z^{4} - 16 = (z^2 + 4)(z^2 - 4)$

STEP 6

Notice that $z^2 - 4$ is also a difference of squares, which can be factored further using the same formula.

STEP 7

Identify $a^2$ and $b^2$ in the expression $z^2 - 4$ . In this case, $a^2 = z^2$ and $b^2 = 4$ .

STEP 8

Find $a$ and $b$ for the new difference of squares.
$a = \sqrt{z^2} = z$ $b = \sqrt{4} = 2$

STEP 9

Apply the difference of squares formula to the expression $z^2 - 4$ .
$z^2 - 4 = (z + 2)(z - 2)$

STEP 10

Combine the factors from Step 5 and Step 9 to express the original expression as a product of factors.
$z^{4} - 16 = (z^2 + 4)(z + 2)(z - 2)$

STEP 11

Check if any further factoring is possible. The term $z^2 + 4$ cannot be factored further using real numbers as it is a sum of squares.

STEP 12

Conclude that the expression $z^{4} - 16$ is factored completely.
$z^{4} - 16 = (z^2 + 4)(z + 2)(z - 2)$

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