Algebra Examples

$f (x) = 3 x^{4} - 2 x^{3} + 74 x^{2} - 50 x - 25$

Step 1

Regroup terms.

$f (x) = - 2 x^{3} - 50 x + 3 x^{4} + 74 x^{2} - 25$

Step 2

Factor $- 2 x$ out of $- 2 x^{3} - 50 x$ .

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Step 2.1

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

$f (x) = - 2 x \cdot x^{2} - 50 x + 3 x^{4} + 74 x^{2} - 25$

Step 2.2

Factor $- 2 x$ out of $- 50 x$ .

$f (x) = - 2 x \cdot x^{2} - 2 x \cdot 25 + 3 x^{4} + 74 x^{2} - 25$

Step 2.3

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

$f (x) = - 2 x (x^{2} + 25) + 3 x^{4} + 74 x^{2} - 25$

Step 3

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

$f (x) = - 2 x (x^{2} + 25) + 3 {(x^{2})}^{2} + 74 x^{2} - 25$

Step 4

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

$f (x) = - 2 x (x^{2} + 25) + 3 u^{2} + 74 u - 25$

Step 5

Factor by grouping.

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Step 5.1

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 25 = - 75$ and whose sum is $i = 74$ .

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Step 5.1.1

Factor $74$ out of $74 u$ .

$f (x) = - 2 x (x^{2} + 25) + 3 u^{2} + 74 (u) - 25$

Step 5.1.2

Rewrite $74$ as $- 1$ plus $75$

$f (x) = - 2 x (x^{2} + 25) + 3 u^{2} + (- 1 + 75) u - 25$

Step 5.1.3

Apply the distributive property.

$f (x) = - 2 x (x^{2} + 25) + 3 u^{2} - 1 u + 75 u - 25$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$f (x) = - 2 x (x^{2} + 25) + 3 u^{2} - 1 u + 75 u - 25$

Step 5.2.2

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

$f (x) = - 2 x (x^{2} + 25) + u (3 u - 1) + 25 (3 u - 1)$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $3 u - 1$ .

$f (x) = - 2 x (x^{2} + 25) + (3 u - 1) (u + 25)$

Step 6

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

$f (x) = - 2 x (x^{2} + 25) + (3 x^{2} - 1) (x^{2} + 25)$

Step 7

Factor $x^{2} + 25$ out of $- 2 x (x^{2} + 25) + (3 x^{2} - 1) (x^{2} + 25)$ .

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Step 7.1

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

$f (x) = (x^{2} + 25) (- 2 x) + (3 x^{2} - 1) (x^{2} + 25)$

Step 7.2

Factor $x^{2} + 25$ out of $(3 x^{2} - 1) (x^{2} + 25)$ .

$f (x) = (x^{2} + 25) (- 2 x) + (x^{2} + 25) (3 x^{2} - 1)$

Step 7.3

Factor $x^{2} + 25$ out of $(x^{2} + 25) (- 2 x) + (x^{2} + 25) (3 x^{2} - 1)$ .

$f (x) = (x^{2} + 25) (- 2 x + 3 x^{2} - 1)$

Step 8

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

$f (x) = (x^{2} + 25) (- 2 u + 3 u^{2} - 1)$

Step 9

Factor by grouping.

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Step 9.1

Reorder terms.

$f (x) = (x^{2} + 25) (3 u^{2} - 2 u - 1)$

Step 9.2

For a polynomial of the form $a x^{2} + i x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $i = - 2$ .

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Step 9.2.1

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

$f (x) = (x^{2} + 25) (3 u^{2} - 2 u - 1)$

Step 9.2.2

Rewrite $- 2$ as $1$ plus $- 3$

$f (x) = (x^{2} + 25) (3 u^{2} + (1 - 3) u - 1)$

Step 9.2.3

Apply the distributive property.

$f (x) = (x^{2} + 25) (3 u^{2} + 1 u - 3 u - 1)$

Step 9.2.4

Multiply $u$ by $1$ .

$f (x) = (x^{2} + 25) (3 u^{2} + u - 3 u - 1)$

Step 9.3

Factor out the greatest common factor from each group.

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Step 9.3.1

Group the first two terms and the last two terms.

$f (x) = (x^{2} + 25) ((3 u^{2} + u) - 3 u - 1)$

Step 9.3.2

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

$f (x) = (x^{2} + 25) (u (3 u + 1) - (3 u + 1))$

Step 9.4

Factor the polynomial by factoring out the greatest common factor, $3 u + 1$ .

$f (x) = (x^{2} + 25) ((3 u + 1) (u - 1))$

Step 10

Factor.

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Step 10.1

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

$f (x) = (x^{2} + 25) ((3 x + 1) (x - 1))$

Step 10.2

Remove unnecessary parentheses.

$f (x) = (x^{2} + 25) (3 x + 1) (x - 1)$