Precalculus Examples

$t^{4} + 2 t^{2} + 1$

Step 1

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

${(t^{2})}^{2} + 2 t^{2} + 1$

Step 2

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

$u^{2} + 2 u + 1$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

$u^{2} + 2 u + 1^{2}$

Step 3.2

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

$2 u = 2 \cdot u \cdot 1$

Step 3.3

Rewrite the polynomial.

$u^{2} + 2 \cdot u \cdot 1 + 1^{2}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

${(u + 1)}^{2}$

Step 4

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

${(t^{2} + 1)}^{2}$