Algebra Examples

$\frac{3 z^{6} + 21 z^{4}}{z^{4} + 11 z^{2} + 28}$

Step 1

Factor $3 z^{4}$ out of $3 z^{6} + 21 z^{4}$ .

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Step 1.1

Factor $3 z^{4}$ out of $3 z^{6}$ .

$\frac{3 z^{4} (z^{2}) + 21 z^{4}}{z^{4} + 11 z^{2} + 28}$

Step 1.2

Factor $3 z^{4}$ out of $21 z^{4}$ .

$\frac{3 z^{4} (z^{2}) + 3 z^{4} (7)}{z^{4} + 11 z^{2} + 28}$

Step 1.3

Factor $3 z^{4}$ out of $3 z^{4} (z^{2}) + 3 z^{4} (7)$ .

$\frac{3 z^{4} (z^{2} + 7)}{z^{4} + 11 z^{2} + 28}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $z^{4}$ as ${(z^{2})}^{2}$ .

$\frac{3 z^{4} (z^{2} + 7)}{{(z^{2})}^{2} + 11 z^{2} + 28}$

Step 2.2

Let $u = z^{2}$ . Substitute $u$ for all occurrences of $z^{2}$ .

$\frac{3 z^{4} (z^{2} + 7)}{u^{2} + 11 u + 28}$

Step 2.3

Factor $u^{2} + 11 u + 28$ using the AC method.

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Step 2.3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $28$ and whose sum is $11$ .

$4, 7$

Step 2.3.2

Write the factored form using these integers.

$\frac{3 z^{4} (z^{2} + 7)}{(u + 4) (u + 7)}$

Step 2.4

Replace all occurrences of $u$ with $z^{2}$ .

$\frac{3 z^{4} (z^{2} + 7)}{(z^{2} + 4) (z^{2} + 7)}$

Step 3

Cancel the common factor of $z^{2} + 7$ .

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Step 3.1

Cancel the common factor.

$\frac{3 z^{4} (z^{2} + 7)}{(z^{2} + 4) (z^{2} + 7)}$

Step 3.2

Rewrite the expression.

$\frac{3 z^{4}}{z^{2} + 4}$