Pre-Algebra Examples

(2 x + 3) (x^{2} - x - 1)

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

Step 1

Expand

(2 x + 3) (x^{2} - x - 1)

$(2 x + 3) (x^{2} - x - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply

x

$x$ by

x^{2}

$x^{2}$ by adding the exponents.

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Step 2.1.1.1

Move

x^{2}

$x^{2}$ .

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

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

Step 2.1.1.2

Multiply

x^{2}

$x^{2}$ by

x

$x$ .

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Step 2.1.1.2.1

Raise

x

$x$ to the power of

1

$1$ .

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

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

Step 2.1.1.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 2.1.1.3

Add

2

$2$ and

1

$1$ .

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

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

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

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

Step 2.1.2

Rewrite using the commutative property of multiplication.

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

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

Step 2.1.3

Multiply

x

$x$ by

x

$x$ by adding the exponents.

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Step 2.1.3.1

Move

x

$x$ .

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

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

Step 2.1.3.2

Multiply

x

$x$ by

x

$x$ .

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

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

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

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

Step 2.1.4

Multiply

2

$2$ by

- 1

$- 1$ .

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

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

Step 2.1.5

Multiply

- 1

$- 1$ by

2

$2$ .

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

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

Step 2.1.6

Multiply

- 1

$- 1$ by

3

$3$ .

2 x^{3} - 2 x^{2} - 2 x + 3 x^{2} - 3 x + 3 \cdot - 1

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

Step 2.1.7

Multiply

3

$3$ by

- 1

$- 1$ .

2 x^{3} - 2 x^{2} - 2 x + 3 x^{2} - 3 x - 3

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

2 x^{3} - 2 x^{2} - 2 x + 3 x^{2} - 3 x - 3

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add

- 2 x^{2}

$- 2 x^{2}$ and

3 x^{2}

$3 x^{2}$ .

2 x^{3} + x^{2} - 2 x - 3 x - 3

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

Step 2.2.2

Subtract

3 x

$3 x$ from

- 2 x

$- 2 x$ .

2 x^{3} + x^{2} - 5 x - 3

$2 x^{3} + x^{2} - 5 x - 3$

2 x^{3} + x^{2} - 5 x - 3

$2 x^{3} + x^{2} - 5 x - 3$

2 x^{3} + x^{2} - 5 x - 3

$2 x^{3} + x^{2} - 5 x - 3$

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