Pre-Algebra Examples

$(16 x^{3} - 9 x - 16 x^{2} + 9) \div (x - 1)$

Step 1

Rewrite the division as a fraction.

$\frac{16 x^{3} - 9 x - 16 x^{2} + 9}{x - 1}$

Step 2

Simplify the numerator.

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Step 2.1

Reorder terms.

$\frac{16 x^{3} - 16 x^{2} - 9 x + 9}{x - 1}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(16 x^{3} - 16 x^{2}) - 9 x + 9}{x - 1}$

Step 2.2.2

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

$\frac{16 x^{2} (x - 1) - 9 (x - 1)}{x - 1}$

Step 2.3

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

$\frac{(x - 1) (16 x^{2} - 9)}{x - 1}$

Step 2.4

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$\frac{(x - 1) ({(4 x)}^{2} - 9)}{x - 1}$

Step 2.5

Rewrite $9$ as $3^{2}$ .

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

Step 2.6

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x$ and $b = 3$ .

$\frac{(x - 1) (4 x + 3) (4 x - 3)}{x - 1}$

Step 3

Cancel the common factor of $x - 1$ .

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Step 3.1

Cancel the common factor.

$\frac{(x - 1) (4 x + 3) (4 x - 3)}{x - 1}$

Step 3.2

Divide $(4 x + 3) (4 x - 3)$ by $1$ .

$(4 x + 3) (4 x - 3)$

Step 4

Expand $(4 x + 3) (4 x - 3)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$4 x (4 x - 3) + 3 (4 x - 3)$

Step 4.2

Apply the distributive property.

$4 x (4 x) + 4 x \cdot - 3 + 3 (4 x - 3)$

Step 4.3

Apply the distributive property.

$4 x (4 x) + 4 x \cdot - 3 + 3 (4 x) + 3 \cdot - 3$

Step 5

Combine the opposite terms in $4 x (4 x) + 4 x \cdot - 3 + 3 (4 x) + 3 \cdot - 3$ .

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Step 5.1

Reorder the factors in the terms $4 x \cdot - 3$ and $3 (4 x)$ .

$4 x (4 x) - 3 \cdot 4 x + 3 \cdot 4 x + 3 \cdot - 3$

Step 5.2

Add $- 3 \cdot 4 x$ and $3 \cdot 4 x$ .

$4 x (4 x) + 0 + 3 \cdot - 3$

Step 5.3

Add $4 x (4 x)$ and $0$ .

$4 x (4 x) + 3 \cdot - 3$

Step 6

Simplify each term.

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Step 6.1

Rewrite using the commutative property of multiplication.

$4 \cdot 4 x \cdot x + 3 \cdot - 3$

Step 6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 6.2.1

Move $x$ .

$4 \cdot 4 (x \cdot x) + 3 \cdot - 3$

Step 6.2.2

Multiply $x$ by $x$ .

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

Step 6.3

Multiply $4$ by $4$ .

$16 x^{2} + 3 \cdot - 3$

Step 6.4

Multiply $3$ by $- 3$ .

$16 x^{2} - 9$