Precalculus Examples

$A^{2} - B^{2} + 16 A + 64$

Step 1

Regroup terms.

$- B^{2} + A^{2} + 16 A + 64$

Step 2

Factor using the perfect square rule.

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Step 2.1

Rewrite $64$ as $8^{2}$ .

$- B^{2} + A^{2} + 16 A + 8^{2}$

Step 2.2

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

$16 A = 2 \cdot A \cdot 8$

Step 2.3

Rewrite the polynomial.

$- B^{2} + A^{2} + 2 \cdot A \cdot 8 + 8^{2}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = A$ and $b = 8$ .

$- B^{2} + {(A + 8)}^{2}$

Step 3

Reorder $- B^{2}$ and ${(A + 8)}^{2}$ .

${(A + 8)}^{2} - B^{2}$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = A + 8$ and $b = B$ .

$(A + 8 + B) (A + 8 - B)$