Precalculus Examples

If $f (x) = 2 x^{3} + 5 x^{2} - 9$ and $g (x) = x^{2} - 2 x + 3$ ; find $(f - g) (x)$

Step 1

Evaluate $f - g$ .

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Step 1.1

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

$(2 x^{3} + 5 x^{2} - 9) - (x^{2} - 2 x + 3)$

Step 1.2

Simplify each term.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Simplify.

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Step 1.2.2.1

Multiply $- 2$ by $- 1$ .

$2 x^{3} + 5 x^{2} - 9 - x^{2} + 2 x - 1 \cdot 3$

Step 1.2.2.2

Multiply $- 1$ by $3$ .

$2 x^{3} + 5 x^{2} - 9 - x^{2} + 2 x - 3$

Step 1.3

Simplify by adding terms.

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Step 1.3.1

Subtract $x^{2}$ from $5 x^{2}$ .

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

Step 1.3.2

Subtract $3$ from $- 9$ .

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

Step 2

Evaluate $f - g$ at $x$ .

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

Step 3

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

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Step 3.1

Remove parentheses.

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

Step 3.2

Multiply $2$ by $x$ .

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