Algebra Examples

$f (x) = x^{4} + x^{3} - 11 x^{2} - 5 x + 30$ , $g (x) = 2 x - 2$

Step 1

Replace the function designators with the actual functions in $f (x) \cdot (g (x))$ .

$(x^{4} + x^{3} - 11 x^{2} - 5 x + 30) \cdot (2 x - 2)$

Step 2

Simplify.

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Step 2.1

Expand $(x^{4} + x^{3} - 11 x^{2} - 5 x + 30) (2 x - 2)$ by multiplying each term in the first expression by each term in the second expression.

$x^{4} (2 x) + x^{4} \cdot - 2 + x^{3} (2 x) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2

Simplify terms.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Rewrite using the commutative property of multiplication.

$2 x^{4} x + x^{4} \cdot - 2 + x^{3} (2 x) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.2

Multiply $x^{4}$ by $x$ by adding the exponents.

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Step 2.2.1.2.1

Move $x$ .

$2 (x \cdot x^{4}) + x^{4} \cdot - 2 + x^{3} (2 x) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.2.2

Multiply $x$ by $x^{4}$ .

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Step 2.2.1.2.2.1

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

$2 (x^{1} x^{4}) + x^{4} \cdot - 2 + x^{3} (2 x) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.2.2.2

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

$2 x^{1 + 4} + x^{4} \cdot - 2 + x^{3} (2 x) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.2.3

Add $1$ and $4$ .

$2 x^{5} + x^{4} \cdot - 2 + x^{3} (2 x) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.3

Move $- 2$ to the left of $x^{4}$ .

$2 x^{5} - 2 \cdot x^{4} + x^{3} (2 x) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.4

Rewrite using the commutative property of multiplication.

$2 x^{5} - 2 x^{4} + 2 x^{3} x + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.5

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 2.2.1.5.1

Move $x$ .

$2 x^{5} - 2 x^{4} + 2 (x \cdot x^{3}) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.5.2

Multiply $x$ by $x^{3}$ .

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Step 2.2.1.5.2.1

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

$2 x^{5} - 2 x^{4} + 2 (x^{1} x^{3}) + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.5.2.2

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

$2 x^{5} - 2 x^{4} + 2 x^{1 + 3} + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.5.3

Add $1$ and $3$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} + x^{3} \cdot - 2 - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.6

Move $- 2$ to the left of $x^{3}$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 \cdot x^{3} - 11 x^{2} (2 x) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.7

Rewrite using the commutative property of multiplication.

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 11 \cdot 2 x^{2} x - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.8

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

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Step 2.2.1.8.1

Move $x$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 11 \cdot 2 (x \cdot x^{2}) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.8.2

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

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Step 2.2.1.8.2.1

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

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 11 \cdot 2 (x^{1} x^{2}) - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.8.2.2

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

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 11 \cdot 2 x^{1 + 2} - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.8.3

Add $1$ and $2$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 11 \cdot 2 x^{3} - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.9

Multiply $- 11$ by $2$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} - 11 x^{2} \cdot - 2 - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.10

Multiply $- 2$ by $- 11$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 5 x (2 x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.11

Rewrite using the commutative property of multiplication.

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 5 \cdot 2 x \cdot x - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.12

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

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Step 2.2.1.12.1

Move $x$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 5 \cdot 2 (x \cdot x) - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.12.2

Multiply $x$ by $x$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 5 \cdot 2 x^{2} - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.13

Multiply $- 5$ by $2$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 10 x^{2} - 5 x \cdot - 2 + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.14

Multiply $- 2$ by $- 5$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 10 x^{2} + 10 x + 30 (2 x) + 30 \cdot - 2$

Step 2.2.1.15

Multiply $2$ by $30$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 10 x^{2} + 10 x + 60 x + 30 \cdot - 2$

Step 2.2.1.16

Multiply $30$ by $- 2$ .

$2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 10 x^{2} + 10 x + 60 x - 60$

Step 2.2.2

Simplify by adding terms.

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Step 2.2.2.1

Combine the opposite terms in $2 x^{5} - 2 x^{4} + 2 x^{4} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 10 x^{2} + 10 x + 60 x - 60$ .

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Step 2.2.2.1.1

Add $- 2 x^{4}$ and $2 x^{4}$ .

$2 x^{5} + 0 - 2 x^{3} - 22 x^{3} + 22 x^{2} - 10 x^{2} + 10 x + 60 x - 60$

Step 2.2.2.1.2

Add $2 x^{5}$ and $0$ .

$2 x^{5} - 2 x^{3} - 22 x^{3} + 22 x^{2} - 10 x^{2} + 10 x + 60 x - 60$

Step 2.2.2.2

Subtract $22 x^{3}$ from $- 2 x^{3}$ .

$2 x^{5} - 24 x^{3} + 22 x^{2} - 10 x^{2} + 10 x + 60 x - 60$

Step 2.2.2.3

Subtract $10 x^{2}$ from $22 x^{2}$ .

$2 x^{5} - 24 x^{3} + 12 x^{2} + 10 x + 60 x - 60$

Step 2.2.2.4

Add $10 x$ and $60 x$ .

$2 x^{5} - 24 x^{3} + 12 x^{2} + 70 x - 60$