Algebra Examples

${(w - 3)}^{5} - {(w - 3)}^{2}$

Step 1

Factor ${(w - 3)}^{2}$ out of ${(w - 3)}^{5} - {(w - 3)}^{2}$ .

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Step 1.1

Factor ${(w - 3)}^{2}$ out of ${(w - 3)}^{5}$ .

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

Step 1.2

Factor ${(w - 3)}^{2}$ out of $- {(w - 3)}^{2}$ .

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

Step 1.3

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

${(w - 3)}^{2} ({(w - 3)}^{3} - 1)$

Step 2

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

${(w - 3)}^{2} ({(w - 3)}^{3} - 1^{3})$

Step 3

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 $a = w - 3$ and $b = 1$ .

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

Step 4

Factor.

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Step 4.1

Simplify.

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Step 4.1.1

Subtract $1$ from $- 3$ .

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

Step 4.1.2

Rewrite ${(w - 3)}^{2}$ as $(w - 3) (w - 3)$ .

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

Step 4.1.3

Expand $(w - 3) (w - 3)$ using the FOIL Method.

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Step 4.1.3.1

Apply the distributive property.

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

Step 4.1.3.2

Apply the distributive property.

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

Step 4.1.3.3

Apply the distributive property.

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

Step 4.1.4

Simplify and combine like terms.

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Step 4.1.4.1

Simplify each term.

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Step 4.1.4.1.1

Multiply $w$ by $w$ .

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

Step 4.1.4.1.2

Move $- 3$ to the left of $w$ .

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

Step 4.1.4.1.3

Multiply $- 3$ by $- 3$ .

${(w - 3)}^{2} ((w - 4) (w^{2} - 3 w - 3 w + 9 + (w - 3) \cdot 1 + 1^{2}))$

Step 4.1.4.2

Subtract $3 w$ from $- 3 w$ .

${(w - 3)}^{2} ((w - 4) (w^{2} - 6 w + 9 + (w - 3) \cdot 1 + 1^{2}))$

Step 4.1.5

Multiply $w - 3$ by $1$ .

${(w - 3)}^{2} ((w - 4) (w^{2} - 6 w + 9 + w - 3 + 1^{2}))$

Step 4.1.6

One to any power is one.

${(w - 3)}^{2} ((w - 4) (w^{2} - 6 w + 9 + w - 3 + 1))$

Step 4.1.7

Add $- 6 w$ and $w$ .

${(w - 3)}^{2} ((w - 4) (w^{2} - 5 w + 9 - 3 + 1))$

Step 4.1.8

Subtract $3$ from $9$ .

${(w - 3)}^{2} ((w - 4) (w^{2} - 5 w + 6 + 1))$

Step 4.1.9

Add $6$ and $1$ .

${(w - 3)}^{2} ((w - 4) (w^{2} - 5 w + 7))$

Step 4.2

Remove unnecessary parentheses.

${(w - 3)}^{2} (w - 4) (w^{2} - 5 w + 7)$