Algebra Examples

$x^{6} - 54 x^{3} + 729$

Step 1

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

${(x^{3})}^{2} - 54 x^{3} + 729$

Step 2

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

$u^{2} - 54 u + 729$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $729$ as $27^{2}$ .

$u^{2} - 54 u + 27^{2}$

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.

$54 u = 2 \cdot u \cdot 27$

Step 3.3

Rewrite the polynomial.

$u^{2} - 2 \cdot u \cdot 27 + 27^{2}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = u$ and $b = 27$ .

${(u - 27)}^{2}$

Step 4

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

${(x^{3} - 27)}^{2}$

Step 5

Rewrite $27$ as $3^{3}$ .

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

Step 6

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 $a = x$ and $b = 3$ .

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

Step 7

Simplify.

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Step 7.1

Move $3$ to the left of $x$ .

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

Step 7.2

Raise $3$ to the power of $2$ .

${((x - 3) (x^{2} + 3 x + 9))}^{2}$

Step 8

Apply the product rule to $(x - 3) (x^{2} + 3 x + 9)$ .

${(x - 3)}^{2} {(x^{2} + 3 x + 9)}^{2}$