Algebra Examples

$\frac{y^{4} + 5 y^{2} + 6}{4 y^{5} + 12 y^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{{(y^{2})}^{2} + 5 y^{2} + 6}{4 y^{5} + 12 y^{3}}$

Step 1.2

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

$\frac{u^{2} + 5 u + 6}{4 y^{5} + 12 y^{3}}$

Step 1.3

Factor $u^{2} + 5 u + 6$ using the AC method.

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Step 1.3.1

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 $6$ and whose sum is $5$ .

$2, 3$

Step 1.3.2

Write the factored form using these integers.

$\frac{(u + 2) (u + 3)}{4 y^{5} + 12 y^{3}}$

Step 1.4

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

$\frac{(y^{2} + 2) (y^{2} + 3)}{4 y^{5} + 12 y^{3}}$

Step 2

Simplify terms.

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Step 2.1

Factor $4 y^{3}$ out of $4 y^{5} + 12 y^{3}$ .

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Step 2.1.1

Factor $4 y^{3}$ out of $4 y^{5}$ .

$\frac{(y^{2} + 2) (y^{2} + 3)}{4 y^{3} (y^{2}) + 12 y^{3}}$

Step 2.1.2

Factor $4 y^{3}$ out of $12 y^{3}$ .

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

Step 2.1.3

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

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

Step 2.2

Cancel the common factor of $y^{2} + 3$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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