Precalculus Examples

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

Step 1

Apply the distributive property.

$4 m^{2} m + 4 m^{2} \cdot - 1 - m + 1$

Step 2

Multiply $m^{2}$ by $m$ by adding the exponents.

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Step 2.1

Move $m$ .

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

Step 2.2

Multiply $m$ by $m^{2}$ .

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Step 2.2.1

Raise $m$ to the power of $1$ .

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

Step 2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$4 m^{1 + 2} + 4 m^{2} \cdot - 1 - m + 1$

Step 2.3

Add $1$ and $2$ .

$4 m^{3} + 4 m^{2} \cdot - 1 - m + 1$

Step 3

Multiply $- 1$ by $4$ .

$4 m^{3} - 4 m^{2} - m + 1$

Step 4

Rewrite $4 m^{3} - 4 m^{2} - m + 1$ in a factored form.

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Step 4.1

Factor out the greatest common factor from each group.

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Step 4.1.1

Group the first two terms and the last two terms.

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

Step 4.1.2

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

$4 m^{2} (m - 1) - (m - 1)$

Step 4.2

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

$(m - 1) (4 m^{2} - 1)$

Step 4.3

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

$(m - 1) ({(2 m)}^{2} - 1)$

Step 4.4

Rewrite $1$ as $1^{2}$ .

$(m - 1) ({(2 m)}^{2} - 1^{2})$

Step 4.5

Factor.

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Step 4.5.1

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

$(m - 1) ((2 m + 1) (2 m - 1))$

Step 4.5.2

Remove unnecessary parentheses.

$(m - 1) (2 m + 1) (2 m - 1)$