Finite Math Examples

$4 y^{4} - 12 y^{2} + 9$

Step 1

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

$4 {(y^{2})}^{2} - 12 y^{2} + 9$

Step 2

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

$4 u^{2} - 12 u + 9$

Step 3

Factor using the perfect square rule.

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Step 3.1

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

${(2 u)}^{2} - 12 u + 9$

Step 3.2

Rewrite $9$ as $3^{2}$ .

${(2 u)}^{2} - 12 u + 3^{2}$

Step 3.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$12 u = 2 \cdot (2 u) \cdot 3$

Step 3.4

Rewrite the polynomial.

${(2 u)}^{2} - 2 \cdot (2 u) \cdot 3 + 3^{2}$

Step 3.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 u$ and $b = 3$ .

${(2 u - 3)}^{2}$

Step 4

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

${(2 y^{2} - 3)}^{2}$