Precalculus Examples

$\cos^{2} (x) + 4 \cos (x) + 3$

Step 1

Let $u = \cos (x)$ . Substitute $u$ for all occurrences of $\cos (x)$ .

$u^{2} + 4 u + 3$

Step 2

Factor $u^{2} + 4 u + 3$ using the AC method.

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Step 2.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 2.2

Write the factored form using these integers.

$(u + 1) (u + 3)$

Step 3

Replace all occurrences of $u$ with $\cos (x)$ .

$(\cos (x) + 1) (\cos (x) + 3)$