Pre-Algebra Examples

$x^{\frac{7}{3}} - x^{\frac{4}{3}} - 2 x^{\frac{1}{3}}$

Step 1

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{7}{3}} - x^{\frac{4}{3}} - 2 x^{\frac{1}{3}}$ .

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Step 1.1

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{7}{3}}$ .

$x^{\frac{1}{3}} x^{\frac{6}{3}} - x^{\frac{4}{3}} - 2 x^{\frac{1}{3}}$

Step 1.2

Factor $x^{\frac{1}{3}}$ out of $- x^{\frac{4}{3}}$ .

$x^{\frac{1}{3}} x^{\frac{6}{3}} + x^{\frac{1}{3}} (- x^{\frac{3}{3}}) - 2 x^{\frac{1}{3}}$

Step 1.3

Factor $x^{\frac{1}{3}}$ out of $- 2 x^{\frac{1}{3}}$ .

$x^{\frac{1}{3}} x^{\frac{6}{3}} + x^{\frac{1}{3}} (- x^{\frac{3}{3}}) + x^{\frac{1}{3}} \cdot - 2$

Step 1.4

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} x^{\frac{6}{3}} + x^{\frac{1}{3}} (- x^{\frac{3}{3}})$ .

$x^{\frac{1}{3}} (x^{\frac{6}{3}} - x^{\frac{3}{3}}) + x^{\frac{1}{3}} \cdot - 2$

Step 1.5

Factor $x^{\frac{1}{3}}$ out of $x^{\frac{1}{3}} (x^{\frac{6}{3}} - x^{\frac{3}{3}}) + x^{\frac{1}{3}} \cdot - 2$ .

$x^{\frac{1}{3}} (x^{\frac{6}{3}} - x^{\frac{3}{3}} - 2)$

Step 2

Factor $x^{\frac{6}{3}} - x^{\frac{3}{3}} - 2$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 2.2

Write the factored form using these integers.

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

Step 3

Divide $3$ by $3$ .

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

Step 4

Simplify.

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

Step 5

Rewrite $x^{\frac{3}{3}}$ as ${(x^{\frac{1}{3}})}^{3}$ .

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

Step 6

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

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

Step 7

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 $a = x^{\frac{1}{3}}$ and $b = 1$ .

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

Step 8

Factor.

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Step 8.1

Factor.

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Step 8.1.1

Simplify.

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Step 8.1.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{2}$ .

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Step 8.1.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 8.1.1.1.2

Combine $\frac{1}{3}$ and $2$ .

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

Step 8.1.1.2

Multiply $- 1$ by $1$ .

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

Step 8.1.1.3

One to any power is one.

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

Step 8.1.2

Remove unnecessary parentheses.

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

Step 8.2

Remove unnecessary parentheses.

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