Algebra Examples

$- 45 x^{4} + 42 x^{3} + 1272 x^{2} - 1050 x - 3675$

Step 1

Factor $3$ out of $- 45 x^{4} + 42 x^{3} + 1272 x^{2} - 1050 x - 3675$ .

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Step 1.1

Factor $3$ out of $- 45 x^{4}$ .

$3 (- 15 x^{4}) + 42 x^{3} + 1272 x^{2} - 1050 x - 3675$

Step 1.2

Factor $3$ out of $42 x^{3}$ .

$3 (- 15 x^{4}) + 3 (14 x^{3}) + 1272 x^{2} - 1050 x - 3675$

Step 1.3

Factor $3$ out of $1272 x^{2}$ .

$3 (- 15 x^{4}) + 3 (14 x^{3}) + 3 (424 x^{2}) - 1050 x - 3675$

Step 1.4

Factor $3$ out of $- 1050 x$ .

$3 (- 15 x^{4}) + 3 (14 x^{3}) + 3 (424 x^{2}) + 3 (- 350 x) - 3675$

Step 1.5

Factor $3$ out of $- 3675$ .

$3 (- 15 x^{4}) + 3 (14 x^{3}) + 3 (424 x^{2}) + 3 (- 350 x) + 3 (- 1225)$

Step 1.6

Factor $3$ out of $3 (- 15 x^{4}) + 3 (14 x^{3})$ .

$3 (- 15 x^{4} + 14 x^{3}) + 3 (424 x^{2}) + 3 (- 350 x) + 3 (- 1225)$

Step 1.7

Factor $3$ out of $3 (- 15 x^{4} + 14 x^{3}) + 3 (424 x^{2})$ .

$3 (- 15 x^{4} + 14 x^{3} + 424 x^{2}) + 3 (- 350 x) + 3 (- 1225)$

Step 1.8

Factor $3$ out of $3 (- 15 x^{4} + 14 x^{3} + 424 x^{2}) + 3 (- 350 x)$ .

$3 (- 15 x^{4} + 14 x^{3} + 424 x^{2} - 350 x) + 3 (- 1225)$

Step 1.9

Factor $3$ out of $3 (- 15 x^{4} + 14 x^{3} + 424 x^{2} - 350 x) + 3 (- 1225)$ .

$3 (- 15 x^{4} + 14 x^{3} + 424 x^{2} - 350 x - 1225)$

Step 2

Regroup terms.

$3 (14 x^{3} - 350 x - 15 x^{4} + 424 x^{2} - 1225)$

Step 3

Factor $14 x$ out of $14 x^{3} - 350 x$ .

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Step 3.1

Factor $14 x$ out of $14 x^{3}$ .

$3 (14 x (x^{2}) - 350 x - 15 x^{4} + 424 x^{2} - 1225)$

Step 3.2

Factor $14 x$ out of $- 350 x$ .

$3 (14 x (x^{2}) + 14 x (- 25) - 15 x^{4} + 424 x^{2} - 1225)$

Step 3.3

Factor $14 x$ out of $14 x (x^{2}) + 14 x (- 25)$ .

$3 (14 x (x^{2} - 25) - 15 x^{4} + 424 x^{2} - 1225)$

Step 4

Rewrite $25$ as $5^{2}$ .

$3 (14 x (x^{2} - 5^{2}) - 15 x^{4} + 424 x^{2} - 1225)$

Step 5

Factor.

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Step 5.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$3 (14 x ((x + 5) (x - 5)) - 15 x^{4} + 424 x^{2} - 1225)$

Step 5.2

Remove unnecessary parentheses.

$3 (14 x (x + 5) (x - 5) - 15 x^{4} + 424 x^{2} - 1225)$

Step 6

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

$3 (14 x (x + 5) (x - 5) - 15 {(x^{2})}^{2} + 424 x^{2} - 1225)$

Step 7

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

$3 (14 x (x + 5) (x - 5) - 15 u^{2} + 424 u - 1225)$

Step 8

Factor by grouping.

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Step 8.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 15 \cdot - 1225 = 18375$ and whose sum is $b = 424$ .

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Step 8.1.1

Factor $424$ out of $424 u$ .

$3 (14 x (x + 5) (x - 5) - 15 u^{2} + 424 (u) - 1225)$

Step 8.1.2

Rewrite $424$ as $49$ plus $375$

$3 (14 x (x + 5) (x - 5) - 15 u^{2} + (49 + 375) u - 1225)$

Step 8.1.3

Apply the distributive property.

$3 (14 x (x + 5) (x - 5) - 15 u^{2} + 49 u + 375 u - 1225)$

Step 8.2

Factor out the greatest common factor from each group.

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Step 8.2.1

Group the first two terms and the last two terms.

$3 (14 x (x + 5) (x - 5) + (- 15 u^{2} + 49 u) + 375 u - 1225)$

Step 8.2.2

Factor out the greatest common factor (GCF) from each group.

$3 (14 x (x + 5) (x - 5) + u (- 15 u + 49) - 25 (- 15 u + 49))$

Step 8.3

Factor the polynomial by factoring out the greatest common factor, $- 15 u + 49$ .

$3 (14 x (x + 5) (x - 5) + (- 15 u + 49) (u - 25))$

Step 9

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

$3 (14 x (x + 5) (x - 5) + (- 15 x^{2} + 49) (x^{2} - 25))$

Step 10

Rewrite $25$ as $5^{2}$ .

$3 (14 x (x + 5) (x - 5) + (- 15 x^{2} + 49) (x^{2} - 5^{2}))$

Step 11

Factor.

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Step 11.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$3 (14 x (x + 5) (x - 5) + (- 15 x^{2} + 49) ((x + 5) (x - 5)))$

Step 11.2

Remove unnecessary parentheses.

$3 (14 x (x + 5) (x - 5) + (- 15 x^{2} + 49) (x + 5) (x - 5))$

Step 12

Factor $(x + 5) (x - 5)$ out of $14 x (x + 5) (x - 5) + (- 15 x^{2} + 49) (x + 5) (x - 5)$ .

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Step 12.1

Factor $(x + 5) (x - 5)$ out of $14 x (x + 5) (x - 5)$ .

$3 ((x + 5) (x - 5) (14 x) + (- 15 x^{2} + 49) (x + 5) (x - 5))$

Step 12.2

Factor $(x + 5) (x - 5)$ out of $(- 15 x^{2} + 49) (x + 5) (x - 5)$ .

$3 ((x + 5) (x - 5) (14 x) + (x + 5) (x - 5) (- 15 x^{2} + 49))$

Step 12.3

Factor $(x + 5) (x - 5)$ out of $(x + 5) (x - 5) (14 x) + (x + 5) (x - 5) (- 15 x^{2} + 49)$ .

$3 ((x + 5) (x - 5) (14 x - 15 x^{2} + 49))$

Step 13

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$3 ((x + 5) (x - 5) (14 u - 15 u^{2} + 49))$

Step 14

Factor by grouping.

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Step 14.1

Reorder terms.

$3 ((x + 5) (x - 5) (- 15 u^{2} + 14 u + 49))$

Step 14.2

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 15 \cdot 49 = - 735$ and whose sum is $b = 14$ .

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Step 14.2.1

Factor $14$ out of $14 u$ .

$3 ((x + 5) (x - 5) (- 15 u^{2} + 14 (u) + 49))$

Step 14.2.2

Rewrite $14$ as $- 21$ plus $35$

$3 ((x + 5) (x - 5) (- 15 u^{2} + (- 21 + 35) u + 49))$

Step 14.2.3

Apply the distributive property.

$3 ((x + 5) (x - 5) (- 15 u^{2} - 21 u + 35 u + 49))$

Step 14.3

Factor out the greatest common factor from each group.

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Step 14.3.1

Group the first two terms and the last two terms.

$3 ((x + 5) (x - 5) ((- 15 u^{2} - 21 u) + 35 u + 49))$

Step 14.3.2

Factor out the greatest common factor (GCF) from each group.

$3 ((x + 5) (x - 5) (3 u (- 5 u - 7) - 7 (- 5 u - 7)))$

Step 14.4

Factor the polynomial by factoring out the greatest common factor, $- 5 u - 7$ .

$3 ((x + 5) (x - 5) ((- 5 u - 7) (3 u - 7)))$

Step 15

Factor.

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Step 15.1

Factor.

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Step 15.1.1

Replace all occurrences of $u$ with $x$ .

$3 ((x + 5) (x - 5) ((- 5 x - 7) (3 x - 7)))$

Step 15.1.2

Remove unnecessary parentheses.

$3 ((x + 5) (x - 5) (- 5 x - 7) (3 x - 7))$

Step 15.2

Remove unnecessary parentheses.

$3 (x + 5) (x - 5) (- 5 x - 7) (3 x - 7)$

Step 16

Factor.

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Step 16.1

Factor $- 1$ out of $- 5 x - 7$ .

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Step 16.1.1

Factor $- 1$ out of $- 5 x$ .

$3 (x + 5) (x - 5) (- (5 x) - 7) (3 x - 7)$

Step 16.1.2

Rewrite $- 7$ as $- 1 (7)$ .

$3 (x + 5) (x - 5) (- (5 x) - 1 (7)) (3 x - 7)$

Step 16.1.3

Factor $- 1$ out of $- (5 x) - 1 (7)$ .

$3 (x + 5) (x - 5) (- (5 x + 7)) (3 x - 7)$

Step 16.2

Remove unnecessary parentheses.

$3 (x + 5) (x - 5) \cdot - 1 (5 x + 7) (3 x - 7)$

Step 17

Factor.

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Step 17.1

Combine exponents.

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Step 17.1.1

Factor out negative.

$- ((3 (x + 5) (x - 5) (5 x + 7)) (3 x - 7))$

Step 17.1.2

Multiply $3$ by $- 1$ .

$- 3 ((((x + 5) (x - 5)) (5 x + 7)) (3 x - 7))$

Step 17.2

Remove unnecessary parentheses.

$- 3 (x + 5) (x - 5) (5 x + 7) (3 x - 7)$