Algebra Examples

$8 x^{6} + 7 x^{3} - 1$

Step 1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

$8 {(x^{3})}^{2} + 7 x^{3} - 1$

Step 2

Let $u = x^{3}$ . Substitute $u$ for all occurrences of $x^{3}$ .

$8 u^{2} + 7 u - 1$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 8 \cdot - 1 = - 8$ and whose sum is $b = 7$ .

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Step 3.1.1

Factor $7$ out of $7 u$ .

$8 u^{2} + 7 (u) - 1$

Step 3.1.2

Rewrite $7$ as $- 1$ plus $8$

$8 u^{2} + (- 1 + 8) u - 1$

Step 3.1.3

Apply the distributive property.

$8 u^{2} - 1 u + 8 u - 1$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(8 u^{2} - 1 u) + 8 u - 1$

Step 3.2.2

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

$u (8 u - 1) + 1 (8 u - 1)$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $8 u - 1$ .

$(8 u - 1) (u + 1)$

Step 4

Replace all occurrences of $u$ with $x^{3}$ .

$(8 x^{3} - 1) (x^{3} + 1)$

Step 5

Rewrite $8 x^{3}$ as ${(2 x)}^{3}$ .

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

Step 6

Rewrite $1$ as $1^{3}$ .

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

Step 7

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 2 x$ and $b = 1$ .

$(2 x - 1) ({(2 x)}^{2} + 2 x \cdot 1 + 1^{2}) (x^{3} + 1)$

Step 8

Simplify.

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Step 8.1

Apply the product rule to $2 x$ .

$(2 x - 1) (2^{2} x^{2} + 2 x \cdot 1 + 1^{2}) (x^{3} + 1)$

Step 8.2

Raise $2$ to the power of $2$ .

$(2 x - 1) (4 x^{2} + 2 x \cdot 1 + 1^{2}) (x^{3} + 1)$

Step 8.3

Multiply $2$ by $1$ .

$(2 x - 1) (4 x^{2} + 2 x + 1^{2}) (x^{3} + 1)$

Step 8.4

One to any power is one.

$(2 x - 1) (4 x^{2} + 2 x + 1) (x^{3} + 1)$

Step 9

Rewrite $1$ as $1^{3}$ .

$(2 x - 1) (4 x^{2} + 2 x + 1) (x^{3} + 1^{3})$

Step 10

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$(2 x - 1) (4 x^{2} + 2 x + 1) ((x + 1) (x^{2} - x \cdot 1 + 1^{2}))$

Step 11

Factor.

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Step 11.1

Simplify.

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Step 11.1.1

Multiply $- 1$ by $1$ .

$(2 x - 1) (4 x^{2} + 2 x + 1) ((x + 1) (x^{2} - x + 1^{2}))$

Step 11.1.2

One to any power is one.

$(2 x - 1) (4 x^{2} + 2 x + 1) ((x + 1) (x^{2} - x + 1))$

Step 11.2

Remove unnecessary parentheses.

$(2 x - 1) (4 x^{2} + 2 x + 1) (x + 1) (x^{2} - x + 1)$