Precalculus Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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Step 2.1.1.1

Move $x$ .

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

Step 2.1.1.2

Multiply $x$ by $x$ .

$(4 x^{2} + 4 x (- b) + b x + b (- b)) (- x + 3 b)$

Step 2.1.2

Rewrite using the commutative property of multiplication.

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

Step 2.1.3

Multiply $4$ by $- 1$ .

$(4 x^{2} - 4 x b + b x + b (- b)) (- x + 3 b)$

Step 2.1.4

Rewrite using the commutative property of multiplication.

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

Step 2.1.5

Multiply $b$ by $b$ by adding the exponents.

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Step 2.1.5.1

Move $b$ .

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

Step 2.1.5.2

Multiply $b$ by $b$ .

$(4 x^{2} - 4 x b + b x - b^{2}) (- x + 3 b)$

Step 2.2

Add $- 4 x b$ and $b x$ .

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Step 2.2.1

Move $x$ .

$(4 x^{2} - 4 b x + b x - b^{2}) (- x + 3 b)$

Step 2.2.2

Add $- 4 b x$ and $b x$ .

$(4 x^{2} - 3 b x - b^{2}) (- x + 3 b)$

Step 3

Expand $(4 x^{2} - 3 b x - b^{2}) (- x + 3 b)$ by multiplying each term in the first expression by each term in the second expression.

$4 x^{2} (- x) + 4 x^{2} (3 b) - 3 b x (- x) - 3 b x (3 b) - b^{2} (- x) - b^{2} (3 b)$

Step 4

Simplify each term.

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Step 4.1

Rewrite using the commutative property of multiplication.

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

Step 4.2

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

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Step 4.2.1

Move $x$ .

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

Step 4.2.2

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

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Step 4.2.2.1

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

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

Step 4.2.2.2

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

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

Step 4.2.3

Add $1$ and $2$ .

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

Step 4.3

Multiply $4$ by $- 1$ .

$- 4 x^{3} + 4 x^{2} (3 b) - 3 b x (- x) - 3 b x (3 b) - b^{2} (- x) - b^{2} (3 b)$

Step 4.4

Rewrite using the commutative property of multiplication.

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

Step 4.5

Multiply $4$ by $3$ .

$- 4 x^{3} + 12 x^{2} b - 3 b x (- x) - 3 b x (3 b) - b^{2} (- x) - b^{2} (3 b)$

Step 4.6

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

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Step 4.6.1

Move $x$ .

$- 4 x^{3} + 12 x^{2} b - 3 b (x \cdot x) \cdot - 1 - 3 b x (3 b) - b^{2} (- x) - b^{2} (3 b)$

Step 4.6.2

Multiply $x$ by $x$ .

$- 4 x^{3} + 12 x^{2} b - 3 b x^{2} \cdot - 1 - 3 b x (3 b) - b^{2} (- x) - b^{2} (3 b)$

Step 4.7

Multiply $- 1$ by $- 3$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 3 b x (3 b) - b^{2} (- x) - b^{2} (3 b)$

Step 4.8

Multiply $b$ by $b$ by adding the exponents.

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Step 4.8.1

Move $b$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 3 (b \cdot b) x \cdot 3 - b^{2} (- x) - b^{2} (3 b)$

Step 4.8.2

Multiply $b$ by $b$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 3 b^{2} x \cdot 3 - b^{2} (- x) - b^{2} (3 b)$

Step 4.9

Multiply $3$ by $- 3$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x - b^{2} (- x) - b^{2} (3 b)$

Step 4.10

Rewrite using the commutative property of multiplication.

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x - 1 \cdot - 1 b^{2} x - b^{2} (3 b)$

Step 4.11

Multiply $- 1$ by $- 1$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x + 1 b^{2} x - b^{2} (3 b)$

Step 4.12

Multiply $b^{2}$ by $1$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x + b^{2} x - b^{2} (3 b)$

Step 4.13

Rewrite using the commutative property of multiplication.

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x + b^{2} x - 1 \cdot 3 b^{2} b$

Step 4.14

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

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Step 4.14.1

Move $b$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x + b^{2} x - 1 \cdot 3 (b \cdot b^{2})$

Step 4.14.2

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

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Step 4.14.2.1

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

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x + b^{2} x - 1 \cdot 3 (b^{1} b^{2})$

Step 4.14.2.2

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

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x + b^{2} x - 1 \cdot 3 b^{1 + 2}$

Step 4.14.3

Add $1$ and $2$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x + b^{2} x - 1 \cdot 3 b^{3}$

Step 4.15

Multiply $- 1$ by $3$ .

$- 4 x^{3} + 12 x^{2} b + 3 b x^{2} - 9 b^{2} x + b^{2} x - 3 b^{3}$

Step 5

Add $12 x^{2} b$ and $3 b x^{2}$ .

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Step 5.1

Move $x^{2}$ .

$- 4 x^{3} + 12 b x^{2} + 3 b x^{2} - 9 b^{2} x + b^{2} x - 3 b^{3}$

Step 5.2

Add $12 b x^{2}$ and $3 b x^{2}$ .

$- 4 x^{3} + 15 b x^{2} - 9 b^{2} x + b^{2} x - 3 b^{3}$

Step 6

Add $- 9 b^{2} x$ and $b^{2} x$ .

$- 4 x^{3} + 15 b x^{2} - 8 b^{2} x - 3 b^{3}$

Step 7

Simplify the expression.

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Step 7.1

Move $- 4 x^{3}$ .

$15 b x^{2} - 8 b^{2} x - 3 b^{3} - 4 x^{3}$

Step 7.2

Move $15 b x^{2}$ .

$- 8 b^{2} x - 3 b^{3} + 15 b x^{2} - 4 x^{3}$

Step 7.3

Reorder $- 8 b^{2} x$ and $- 3 b^{3}$ .

$- 3 b^{3} - 8 b^{2} x + 15 b x^{2} - 4 x^{3}$