Algebra Examples

$(f g) (2)$ $f (x) = x - 6$ $g (x) = - 3 x^{2} - 7 x + 5$

Step 1

Evaluate $f g$ .

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Step 1.1

Replace the function designators in $f g$ with the actual functions.

$(x - 6) (- 3 x^{2} - 7 x + 5)$

Step 1.2

Expand $(x - 6) (- 3 x^{2} - 7 x + 5)$ by multiplying each term in the first expression by each term in the second expression.

$x (- 3 x^{2}) + x (- 7 x) + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3

Simplify terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$- 3 x \cdot x^{2} + x (- 7 x) + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x^{2}$ .

$- 3 (x^{2} x) + x (- 7 x) + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.2.2

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

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Step 1.3.1.2.2.1

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

$- 3 (x^{2} x^{1}) + x (- 7 x) + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.2.2.2

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

$- 3 x^{2 + 1} + x (- 7 x) + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.2.3

Add $2$ and $1$ .

$- 3 x^{3} + x (- 7 x) + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.3

Rewrite using the commutative property of multiplication.

$- 3 x^{3} - 7 x \cdot x + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.4

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

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Step 1.3.1.4.1

Move $x$ .

$- 3 x^{3} - 7 (x \cdot x) + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.4.2

Multiply $x$ by $x$ .

$- 3 x^{3} - 7 x^{2} + x \cdot 5 - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.5

Move $5$ to the left of $x$ .

$- 3 x^{3} - 7 x^{2} + 5 \cdot x - 6 (- 3 x^{2}) - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.6

Multiply $- 3$ by $- 6$ .

$- 3 x^{3} - 7 x^{2} + 5 x + 18 x^{2} - 6 (- 7 x) - 6 \cdot 5$

Step 1.3.1.7

Multiply $- 7$ by $- 6$ .

$- 3 x^{3} - 7 x^{2} + 5 x + 18 x^{2} + 42 x - 6 \cdot 5$

Step 1.3.1.8

Multiply $- 6$ by $5$ .

$- 3 x^{3} - 7 x^{2} + 5 x + 18 x^{2} + 42 x - 30$

Step 1.3.2

Simplify by adding terms.

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Step 1.3.2.1

Add $- 7 x^{2}$ and $18 x^{2}$ .

$- 3 x^{3} + 11 x^{2} + 5 x + 42 x - 30$

Step 1.3.2.2

Add $5 x$ and $42 x$ .

$- 3 x^{3} + 11 x^{2} + 47 x - 30$

Step 2

Evaluate $f g$ at $2$ .

$- 3 {(2)}^{3} + 11 {(2)}^{2} + 47 (2) - 30$

Step 3

Simplify $- 3 {(2)}^{3} + 11 {(2)}^{2} + 47 (2) - 30$ .

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Step 3.1

Remove parentheses.

$- 3 {(2)}^{3} + 11 {(2)}^{2} + 47 (2) - 30$

Step 3.2

Simplify each term.

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Step 3.2.1

Raise $2$ to the power of $3$ .

$- 3 \cdot 8 + 11 {(2)}^{2} + 47 (2) - 30$

Step 3.2.2

Multiply $- 3$ by $8$ .

$- 24 + 11 {(2)}^{2} + 47 (2) - 30$

Step 3.2.3

Raise $2$ to the power of $2$ .

$- 24 + 11 \cdot 4 + 47 (2) - 30$

Step 3.2.4

Multiply $11$ by $4$ .

$- 24 + 44 + 47 (2) - 30$

Step 3.2.5

Multiply $47$ by $2$ .

$- 24 + 44 + 94 - 30$

Step 3.3

Simplify by adding and subtracting.

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Step 3.3.1

Add $- 24$ and $44$ .

$20 + 94 - 30$

Step 3.3.2

Add $20$ and $94$ .

$114 - 30$

Step 3.3.3

Subtract $30$ from $114$ .

$84$