Precalculus Examples

$4 a^{2} c^{2} - {(a^{2} - b^{2} + c^{2})}^{2}$

Step 1

Rewrite $4 a^{2} c^{2}$ as ${(2 a c)}^{2}$ .

${(2 a c)}^{2} - {(a^{2} - b^{2} + c^{2})}^{2}$

Step 2

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

$(2 a c + a^{2} - b^{2} + c^{2}) (2 a c - (a^{2} - b^{2} + c^{2}))$

Step 3

Simplify.

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Step 3.1

Rewrite $2 a c + a^{2} - b^{2} + c^{2}$ in a factored form.

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Step 3.1.1

Regroup terms.

$(- b^{2} + 2 a c + a^{2} + c^{2}) (2 a c - (a^{2} - b^{2} + c^{2}))$

Step 3.1.2

Factor using the perfect square rule.

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Step 3.1.2.1

Rearrange terms.

$(- b^{2} + a^{2} + 2 a c + c^{2}) (2 a c - (a^{2} - b^{2} + c^{2}))$

Step 3.1.2.2

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

$2 a c = 2 \cdot a \cdot c$

Step 3.1.2.3

Rewrite the polynomial.

$(- b^{2} + a^{2} + 2 \cdot a \cdot c + c^{2}) (2 a c - (a^{2} - b^{2} + c^{2}))$

Step 3.1.2.4

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

$(- b^{2} + {(a + c)}^{2}) (2 a c - (a^{2} - b^{2} + c^{2}))$

Step 3.1.3

Reorder $- b^{2}$ and ${(a + c)}^{2}$ .

$({(a + c)}^{2} - b^{2}) (2 a c - (a^{2} - b^{2} + c^{2}))$

Step 3.1.4

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

$(a + c + b) (a + c - b) (2 a c - (a^{2} - b^{2} + c^{2}))$

Step 3.2

Apply the distributive property.

$(a + c + b) (a + c - b) (2 a c - a^{2} - - b^{2} - c^{2})$

Step 3.3

Multiply $- - b^{2}$ .

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Step 3.3.1

Multiply $- 1$ by $- 1$ .

$(a + c + b) (a + c - b) (2 a c - a^{2} + 1 b^{2} - c^{2})$

Step 3.3.2

Multiply $b^{2}$ by $1$ .

$(a + c + b) (a + c - b) (2 a c - a^{2} + b^{2} - c^{2})$

Step 3.4

Factor.

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Step 3.4.1

Rewrite $2 a c - a^{2} + b^{2} - c^{2}$ in a factored form.

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Step 3.4.1.1

Regroup terms.

$(a + c + b) (a + c - b) (b^{2} + 2 a c - a^{2} - c^{2})$

Step 3.4.1.2

Add parentheses.

$(a + c + b) (a + c - b) (b^{2} + 2 (a c) - a^{2} - c^{2})$

Step 3.4.1.3

Let $u = a c$ . Substitute $u$ for all occurrences of $a c$ .

$(a + c + b) (a + c - b) (b^{2} + 2 u - a^{2} - c^{2})$

Step 3.4.1.4

Factor $- 1$ out of $2 u - a^{2} - c^{2}$ .

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Step 3.4.1.4.1

Move $2 u$ .

$(a + c + b) (a + c - b) (b^{2} - a^{2} - c^{2} + 2 u)$

Step 3.4.1.4.2

Factor $- 1$ out of $- a^{2}$ .

$(a + c + b) (a + c - b) (b^{2} - (a^{2}) - c^{2} + 2 u)$

Step 3.4.1.4.3

Factor $- 1$ out of $- c^{2}$ .

$(a + c + b) (a + c - b) (b^{2} - (a^{2}) - (c^{2}) + 2 u)$

Step 3.4.1.4.4

Factor $- 1$ out of $2 u$ .

$(a + c + b) (a + c - b) (b^{2} - (a^{2}) - (c^{2}) - (- 2 u))$

Step 3.4.1.4.5

Factor $- 1$ out of $- (a^{2}) - (c^{2})$ .

$(a + c + b) (a + c - b) (b^{2} - (a^{2} + c^{2}) - (- 2 u))$

Step 3.4.1.4.6

Factor $- 1$ out of $- (a^{2} + c^{2}) - (- 2 u)$ .

$(a + c + b) (a + c - b) (b^{2} - (a^{2} + c^{2} - 2 u))$

Step 3.4.1.5

Replace all occurrences of $u$ with $a c$ .

$(a + c + b) (a + c - b) (b^{2} - (a^{2} + c^{2} - 2 (a c)))$

Step 3.4.1.6

Remove parentheses.

$(a + c + b) (a + c - b) (b^{2} - (a^{2} + c^{2} - 2 a c))$

Step 3.4.1.7

Factor using the perfect square rule.

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Step 3.4.1.7.1

Rearrange terms.

$(a + c + b) (a + c - b) (b^{2} - (a^{2} - 2 a c + c^{2}))$

Step 3.4.1.7.2

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

$2 a c = 2 \cdot a \cdot c$

Step 3.4.1.7.3

Rewrite the polynomial.

$(a + c + b) (a + c - b) (b^{2} - (a^{2} - 2 \cdot a \cdot c + c^{2}))$

Step 3.4.1.7.4

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

$(a + c + b) (a + c - b) (b^{2} - {(a - c)}^{2})$

Step 3.4.1.8

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

$(a + c + b) (a + c - b) ((b + a - c) (b - (a - c)))$

Step 3.4.1.9

Simplify.

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Step 3.4.1.9.1

Apply the distributive property.

$(a + c + b) (a + c - b) ((b + a - c) (b - a - - c))$

Step 3.4.1.9.2

Multiply $- - c$ .

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Step 3.4.1.9.2.1

Multiply $- 1$ by $- 1$ .

$(a + c + b) (a + c - b) ((b + a - c) (b - a + 1 c))$

Step 3.4.1.9.2.2

Multiply $c$ by $1$ .

$(a + c + b) (a + c - b) ((b + a - c) (b - a + c))$

Step 3.4.2

Remove unnecessary parentheses.

$(a + c + b) (a + c - b) (b + a - c) (b - a + c)$