Algebra Examples

$x^{3} + 2 x^{2} - 23 x - 60 = (x + 3) (x + 4) (x - 5)$

Step 1

Simplify $(x + 3) (x + 4) (x - 5)$ .

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Step 1.1

Rewrite.

$x^{3} + 2 x^{2} - 23 x - 60 = 0 + 0 + (x + 3) (x + 4) (x - 5)$

Step 1.2

Simplify by adding zeros.

$x^{3} + 2 x^{2} - 23 x - 60 = (x + 3) (x + 4) (x - 5)$

Step 1.3

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

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Step 1.3.1

Apply the distributive property.

$x^{3} + 2 x^{2} - 23 x - 60 = (x (x + 4) + 3 (x + 4)) (x - 5)$

Step 1.3.2

Apply the distributive property.

$x^{3} + 2 x^{2} - 23 x - 60 = (x \cdot x + x \cdot 4 + 3 (x + 4)) (x - 5)$

Step 1.3.3

Apply the distributive property.

$x^{3} + 2 x^{2} - 23 x - 60 = (x \cdot x + x \cdot 4 + 3 x + 3 \cdot 4) (x - 5)$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

$x^{3} + 2 x^{2} - 23 x - 60 = (x^{2} + x \cdot 4 + 3 x + 3 \cdot 4) (x - 5)$

Step 1.4.1.2

Move $4$ to the left of $x$ .

$x^{3} + 2 x^{2} - 23 x - 60 = (x^{2} + 4 \cdot x + 3 x + 3 \cdot 4) (x - 5)$

Step 1.4.1.3

Multiply $3$ by $4$ .

$x^{3} + 2 x^{2} - 23 x - 60 = (x^{2} + 4 x + 3 x + 12) (x - 5)$

Step 1.4.2

Add $4 x$ and $3 x$ .

$x^{3} + 2 x^{2} - 23 x - 60 = (x^{2} + 7 x + 12) (x - 5)$

Step 1.5

Expand $(x^{2} + 7 x + 12) (x - 5)$ by multiplying each term in the first expression by each term in the second expression.

$x^{3} + 2 x^{2} - 23 x - 60 = x^{2} x + x^{2} \cdot - 5 + 7 x \cdot x + 7 x \cdot - 5 + 12 x + 12 \cdot - 5$

Step 1.6

Simplify terms.

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Step 1.6.1

Simplify each term.

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Step 1.6.1.1

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

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Step 1.6.1.1.1

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

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Step 1.6.1.1.1.1

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

$x^{3} + 2 x^{2} - 23 x - 60 = x^{2} x^{1} + x^{2} \cdot - 5 + 7 x \cdot x + 7 x \cdot - 5 + 12 x + 12 \cdot - 5$

Step 1.6.1.1.1.2

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

$x^{3} + 2 x^{2} - 23 x - 60 = x^{2 + 1} + x^{2} \cdot - 5 + 7 x \cdot x + 7 x \cdot - 5 + 12 x + 12 \cdot - 5$

Step 1.6.1.1.2

Add $2$ and $1$ .

$x^{3} + 2 x^{2} - 23 x - 60 = x^{3} + x^{2} \cdot - 5 + 7 x \cdot x + 7 x \cdot - 5 + 12 x + 12 \cdot - 5$

Step 1.6.1.2

Move $- 5$ to the left of $x^{2}$ .

$x^{3} + 2 x^{2} - 23 x - 60 = x^{3} - 5 \cdot x^{2} + 7 x \cdot x + 7 x \cdot - 5 + 12 x + 12 \cdot - 5$

Step 1.6.1.3

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

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Step 1.6.1.3.1

Move $x$ .

$x^{3} + 2 x^{2} - 23 x - 60 = x^{3} - 5 x^{2} + 7 (x \cdot x) + 7 x \cdot - 5 + 12 x + 12 \cdot - 5$

Step 1.6.1.3.2

Multiply $x$ by $x$ .

$x^{3} + 2 x^{2} - 23 x - 60 = x^{3} - 5 x^{2} + 7 x^{2} + 7 x \cdot - 5 + 12 x + 12 \cdot - 5$

Step 1.6.1.4

Multiply $- 5$ by $7$ .

$x^{3} + 2 x^{2} - 23 x - 60 = x^{3} - 5 x^{2} + 7 x^{2} - 35 x + 12 x + 12 \cdot - 5$

Step 1.6.1.5

Multiply $12$ by $- 5$ .

$x^{3} + 2 x^{2} - 23 x - 60 = x^{3} - 5 x^{2} + 7 x^{2} - 35 x + 12 x - 60$

Step 1.6.2

Simplify by adding terms.

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Step 1.6.2.1

Add $- 5 x^{2}$ and $7 x^{2}$ .

$x^{3} + 2 x^{2} - 23 x - 60 = x^{3} + 2 x^{2} - 35 x + 12 x - 60$

Step 1.6.2.2

Add $- 35 x$ and $12 x$ .

$x^{3} + 2 x^{2} - 23 x - 60 = x^{3} + 2 x^{2} - 23 x - 60$

Step 2

Move all terms containing $x$ to the left side of the equation.

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Step 2.1

Subtract $x^{3}$ from both sides of the equation.

$x^{3} + 2 x^{2} - 23 x - 60 - x^{3} = 2 x^{2} - 23 x - 60$

Step 2.2

Subtract $2 x^{2}$ from both sides of the equation.

$x^{3} + 2 x^{2} - 23 x - 60 - x^{3} - 2 x^{2} = - 23 x - 60$

Step 2.3

Add $23 x$ to both sides of the equation.

$x^{3} + 2 x^{2} - 23 x - 60 - x^{3} - 2 x^{2} + 23 x = - 60$

Step 2.4

Combine the opposite terms in $x^{3} + 2 x^{2} - 23 x - 60 - x^{3} - 2 x^{2} + 23 x$ .

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Step 2.4.1

Subtract $x^{3}$ from $x^{3}$ .

$2 x^{2} - 23 x - 60 + 0 - 2 x^{2} + 23 x = - 60$

Step 2.4.2

Add $2 x^{2} - 23 x - 60$ and $0$ .

$2 x^{2} - 23 x - 60 - 2 x^{2} + 23 x = - 60$

Step 2.4.3

Subtract $2 x^{2}$ from $2 x^{2}$ .

$- 23 x - 60 + 0 + 23 x = - 60$

Step 2.4.4

Add $- 23 x - 60$ and $0$ .

$- 23 x - 60 + 23 x = - 60$

Step 2.4.5

Add $- 23 x$ and $23 x$ .

$0 - 60 = - 60$

Step 2.4.6

Subtract $60$ from $0$ .

$- 60 = - 60$

Step 3

Since $- 60 = - 60$ , the equation will always be true.

Always true