How do you factor $z^{4} - 10 z^{2} + 9$ $z^4 - 10z^2 + 9$ ?

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How do you factor $z^{4} - 10 z^{2} + 9$ $z^4 - 10z^2 + 9$ ?

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Answer 1 · 2016-09-02T17:52:19

$z^{4} - 10 z^{2} + 9 = (z - 1) (z + 1) (z - 3) (z + 3)$ $z^4-10z^2+9 = (z-1)(z+1)(z-3)(z+3)$

Explanation:

The difference of square identity can be written:

$a^{2} - b^{2} = (a - b) (a + b)$ $a^2-b^2=(a-b)(a+b)$

We will use this a couple of times.

Let us treat this as a quadratic in $z^{2}$ $z^2$ . Note that the sum of the coefficients is $0$ $0$ . That is $1 - 10 + 9 = 0$ $1-10+9 = 0$ . Hence $((z^{2}) - 1)$ $((z^2)-1)$ is a factor and the other factor must be $((z^{2}) - 9)$ $((z^2)-9)$ ...

$z^{4} - 10 z^{2} + 9 = {(z^{2})}^{2} - 10 (z^{2}) - 9$ $z^4-10z^2+9 = (z^2)^2-10(z^2)-9$

$z^{4} - 10 z^{2} + 9 = (z^{2} - 1) (z^{2} - 9)$ $color(white)(z^4-10z^2+9) = (z^2-1)(z^2-9)$

$z^{4} - 10 z^{2} + 9 = (z^{2} - 1^{2}) (z^{2} - 3^{2})$ $color(white)(z^4-10z^2+9) = (z^2-1^2)(z^2-3^2)$

$z^{4} - 10 z^{2} + 9 = (z - 1) (z + 1) (z - 3) (z + 3)$ $color(white)(z^4-10z^2+9) = (z-1)(z+1)(z-3)(z+3)$

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How do you factor $z^{4} - 10 z^{2} + 9$ $z^4 - 10z^2 + 9$ ?

1 Answer

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How do you factor z4−10z2+9z^4 - 10z^2 + 9?

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How do you factor $z^{4} - 10 z^{2} + 9$ $z^4 - 10z^2 + 9$ ?