Math  /  Algebra

QuestionFactor the binomial completely. z41z^{4}-1
Part: 0/40 / 4 \square
Part 1 of 4
The GCF is 1 . z41z^{4}-1 is a difference of squares. Write in the form a2b2a^{2}-b^{2}, where a=z2a=z^{2} and b=b= \square .

Studdy Solution

STEP 1

1. We are asked to factor the binomial z41 z^4 - 1 completely.
2. The expression is a difference of squares, which can be factored using the identity a2b2=(ab)(a+b) a^2 - b^2 = (a - b)(a + b) .

STEP 2

1. Identify the expression as a difference of squares.
2. Apply the difference of squares formula.
3. Factor any remaining expressions further if possible.

STEP 3

Recognize that z41 z^4 - 1 is a difference of squares. We can express it in the form a2b2 a^2 - b^2 .
z41=(z2)212 z^4 - 1 = (z^2)^2 - 1^2
Here, a=z2 a = z^2 and b=1 b = 1 .

STEP 4

Apply the difference of squares formula:
(z2)212=(z21)(z2+1) (z^2)^2 - 1^2 = (z^2 - 1)(z^2 + 1)

STEP 5

Notice that z21 z^2 - 1 is also a difference of squares and can be factored further:
z21=(z1)(z+1) z^2 - 1 = (z - 1)(z + 1)
So, the complete factorization of the original expression is:
(z1)(z+1)(z2+1) (z - 1)(z + 1)(z^2 + 1)

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