Precalculus Examples

Given that $f (x) = x^{2} - 3 x - 54$ and $g (x) = x + 6$ ; find $(f \cdot g) (x)$ and express the result in standard form.

Step 1

Evaluate $f \cdot g$ .

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Step 1.1

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

$(x^{2} - 3 x - 54) \cdot (x + 6)$

Step 1.2

Expand $(x^{2} - 3 x - 54) (x + 6)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x + x^{2} \cdot 6 - 3 x \cdot x - 3 x \cdot 6 - 54 x - 54 \cdot 6$

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

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

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Step 1.3.1.1.1

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

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Step 1.3.1.1.1.1

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

$x^{2} x^{1} + x^{2} \cdot 6 - 3 x \cdot x - 3 x \cdot 6 - 54 x - 54 \cdot 6$

Step 1.3.1.1.1.2

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

$x^{2 + 1} + x^{2} \cdot 6 - 3 x \cdot x - 3 x \cdot 6 - 54 x - 54 \cdot 6$

Step 1.3.1.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot 6 - 3 x \cdot x - 3 x \cdot 6 - 54 x - 54 \cdot 6$

Step 1.3.1.2

Move $6$ to the left of $x^{2}$ .

$x^{3} + 6 \cdot x^{2} - 3 x \cdot x - 3 x \cdot 6 - 54 x - 54 \cdot 6$

Step 1.3.1.3

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

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Step 1.3.1.3.1

Move $x$ .

$x^{3} + 6 x^{2} - 3 (x \cdot x) - 3 x \cdot 6 - 54 x - 54 \cdot 6$

Step 1.3.1.3.2

Multiply $x$ by $x$ .

$x^{3} + 6 x^{2} - 3 x^{2} - 3 x \cdot 6 - 54 x - 54 \cdot 6$

Step 1.3.1.4

Multiply $6$ by $- 3$ .

$x^{3} + 6 x^{2} - 3 x^{2} - 18 x - 54 x - 54 \cdot 6$

Step 1.3.1.5

Multiply $- 54$ by $6$ .

$x^{3} + 6 x^{2} - 3 x^{2} - 18 x - 54 x - 324$

Step 1.3.2

Simplify by adding terms.

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Step 1.3.2.1

Subtract $3 x^{2}$ from $6 x^{2}$ .

$x^{3} + 3 x^{2} - 18 x - 54 x - 324$

Step 1.3.2.2

Subtract $54 x$ from $- 18 x$ .

$x^{3} + 3 x^{2} - 72 x - 324$

Step 2

Evaluate $f \cdot g$ at $x$ .

${(x)}^{3} + 3 {(x)}^{2} - 72 x - 324$

Step 3

Simplify ${(x)}^{3} + 3 {(x)}^{2} - 72 x - 324$ .

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Step 3.1

Remove parentheses.

${(x)}^{3} + 3 {(x)}^{2} - 72 x - 324$

Step 3.2

Remove parentheses.

$x^{3} + 3 x^{2} - 72 x - 324$