Algebra Examples

$x^{3} - 8 x^{2} + 7 x^{2} + 16 x - 56 x + 112$

Step 1

Regroup terms.

$x^{3} - 8 x^{2} + 16 x + 7 x^{2} - 56 x + 112$

Step 2

Factor $x$ out of $x^{3} - 8 x^{2} + 16 x$ .

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Step 2.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - 8 x^{2} + 16 x + 7 x^{2} - 56 x + 112$

Step 2.2

Factor $x$ out of $- 8 x^{2}$ .

$x \cdot x^{2} + x (- 8 x) + 16 x + 7 x^{2} - 56 x + 112$

Step 2.3

Factor $x$ out of $16 x$ .

$x \cdot x^{2} + x (- 8 x) + x \cdot 16 + 7 x^{2} - 56 x + 112$

Step 2.4

Factor $x$ out of $x \cdot x^{2} + x (- 8 x)$ .

$x (x^{2} - 8 x) + x \cdot 16 + 7 x^{2} - 56 x + 112$

Step 2.5

Factor $x$ out of $x (x^{2} - 8 x) + x \cdot 16$ .

$x (x^{2} - 8 x + 16) + 7 x^{2} - 56 x + 112$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $16$ as $4^{2}$ .

$x (x^{2} - 8 x + 4^{2}) + 7 x^{2} - 56 x + 112$

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.

$8 x = 2 \cdot x \cdot 4$

Step 3.3

Rewrite the polynomial.

$x (x^{2} - 2 \cdot x \cdot 4 + 4^{2}) + 7 x^{2} - 56 x + 112$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

$x {(x - 4)}^{2} + 7 x^{2} - 56 x + 112$

Step 4

Factor $7$ out of $7 x^{2} - 56 x + 112$ .

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Step 4.1

Factor $7$ out of $7 x^{2}$ .

$x {(x - 4)}^{2} + 7 (x^{2}) - 56 x + 112$

Step 4.2

Factor $7$ out of $- 56 x$ .

$x {(x - 4)}^{2} + 7 (x^{2}) + 7 (- 8 x) + 112$

Step 4.3

Factor $7$ out of $112$ .

$x {(x - 4)}^{2} + 7 x^{2} + 7 (- 8 x) + 7 \cdot 16$

Step 4.4

Factor $7$ out of $7 x^{2} + 7 (- 8 x)$ .

$x {(x - 4)}^{2} + 7 (x^{2} - 8 x) + 7 \cdot 16$

Step 4.5

Factor $7$ out of $7 (x^{2} - 8 x) + 7 \cdot 16$ .

$x {(x - 4)}^{2} + 7 (x^{2} - 8 x + 16)$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $16$ as $4^{2}$ .

$x {(x - 4)}^{2} + 7 (x^{2} - 8 x + 4^{2})$

Step 5.2

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

$8 x = 2 \cdot x \cdot 4$

Step 5.3

Rewrite the polynomial.

$x {(x - 4)}^{2} + 7 (x^{2} - 2 \cdot x \cdot 4 + 4^{2})$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 4$ .

$x {(x - 4)}^{2} + 7 {(x - 4)}^{2}$

Step 6

Factor ${(x - 4)}^{2}$ out of $x {(x - 4)}^{2} + 7 {(x - 4)}^{2}$ .

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Step 6.1

Factor ${(x - 4)}^{2}$ out of $x {(x - 4)}^{2}$ .

${(x - 4)}^{2} x + 7 {(x - 4)}^{2}$

Step 6.2

Factor ${(x - 4)}^{2}$ out of $7 {(x - 4)}^{2}$ .

${(x - 4)}^{2} x + {(x - 4)}^{2} \cdot 7$

Step 6.3

Factor ${(x - 4)}^{2}$ out of ${(x - 4)}^{2} x + {(x - 4)}^{2} \cdot 7$ .

${(x - 4)}^{2} (x + 7)$