Precalculus Examples

Find $(g \cdot h) (x)$ if $g (x) = x^{2} + 4 x + 9$ and $h (x) = x^{2} - 5 x + 10$

Step 1

Evaluate $g \cdot h$ .

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Step 1.1

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

$(x^{2} + 4 x + 9) \cdot (x^{2} - 5 x + 10)$

Step 1.2

Expand $(x^{2} + 4 x + 9) (x^{2} - 5 x + 10)$ by multiplying each term in the first expression by each term in the second expression.

$x^{2} x^{2} + x^{2} (- 5 x) + x^{2} \cdot 10 + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

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

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Step 1.3.1.1.1

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

$x^{2 + 2} + x^{2} (- 5 x) + x^{2} \cdot 10 + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.1.2

Add $2$ and $2$ .

$x^{4} + x^{2} (- 5 x) + x^{2} \cdot 10 + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.2

Rewrite using the commutative property of multiplication.

$x^{4} - 5 x^{2} x + x^{2} \cdot 10 + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.3

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

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Step 1.3.1.3.1

Move $x$ .

$x^{4} - 5 (x \cdot x^{2}) + x^{2} \cdot 10 + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.3.2

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

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Step 1.3.1.3.2.1

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

$x^{4} - 5 (x^{1} x^{2}) + x^{2} \cdot 10 + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.3.2.2

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

$x^{4} - 5 x^{1 + 2} + x^{2} \cdot 10 + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.3.3

Add $1$ and $2$ .

$x^{4} - 5 x^{3} + x^{2} \cdot 10 + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.4

Move $10$ to the left of $x^{2}$ .

$x^{4} - 5 x^{3} + 10 \cdot x^{2} + 4 x \cdot x^{2} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.5

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

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Step 1.3.1.5.1

Move $x^{2}$ .

$x^{4} - 5 x^{3} + 10 x^{2} + 4 (x^{2} x) + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.5.2

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

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Step 1.3.1.5.2.1

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

$x^{4} - 5 x^{3} + 10 x^{2} + 4 (x^{2} x^{1}) + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.5.2.2

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

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{2 + 1} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.5.3

Add $2$ and $1$ .

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{3} + 4 x (- 5 x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.6

Rewrite using the commutative property of multiplication.

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{3} + 4 \cdot - 5 x \cdot x + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.7

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

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Step 1.3.1.7.1

Move $x$ .

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{3} + 4 \cdot - 5 (x \cdot x) + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.7.2

Multiply $x$ by $x$ .

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{3} + 4 \cdot - 5 x^{2} + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.8

Multiply $4$ by $- 5$ .

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{3} - 20 x^{2} + 4 x \cdot 10 + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.9

Multiply $10$ by $4$ .

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{3} - 20 x^{2} + 40 x + 9 x^{2} + 9 (- 5 x) + 9 \cdot 10$

Step 1.3.1.10

Multiply $- 5$ by $9$ .

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{3} - 20 x^{2} + 40 x + 9 x^{2} - 45 x + 9 \cdot 10$

Step 1.3.1.11

Multiply $9$ by $10$ .

$x^{4} - 5 x^{3} + 10 x^{2} + 4 x^{3} - 20 x^{2} + 40 x + 9 x^{2} - 45 x + 90$

Step 1.3.2

Simplify by adding terms.

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Step 1.3.2.1

Add $- 5 x^{3}$ and $4 x^{3}$ .

$x^{4} - x^{3} + 10 x^{2} - 20 x^{2} + 40 x + 9 x^{2} - 45 x + 90$

Step 1.3.2.2

Subtract $20 x^{2}$ from $10 x^{2}$ .

$x^{4} - x^{3} - 10 x^{2} + 40 x + 9 x^{2} - 45 x + 90$

Step 1.3.2.3

Add $- 10 x^{2}$ and $9 x^{2}$ .

$x^{4} - x^{3} - x^{2} + 40 x - 45 x + 90$

Step 1.3.2.4

Subtract $45 x$ from $40 x$ .

$x^{4} - x^{3} - x^{2} - 5 x + 90$

Step 2

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

${(x)}^{4} - {(x)}^{3} - {(x)}^{2} - 5 x + 90$

Step 3

Simplify ${(x)}^{4} - {(x)}^{3} - {(x)}^{2} - 5 x + 90$ .

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Step 3.1

Remove parentheses.

${(x)}^{4} - {(x)}^{3} - {(x)}^{2} - 5 x + 90$

Step 3.2

Remove parentheses.

$x^{4} - x^{3} - x^{2} - 5 x + 90$