Precalculus Examples

$t = (x - 5) (x^{2} + x) + (x - 5)$

Step 1

Remove parentheses.

$t = (x - 5) (x^{2} + x) + x - 5$

Step 2

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

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Step 2.1

Simplify each term.

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Step 2.1.1

Expand $(x - 5) (x^{2} + x)$ using the FOIL Method.

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Step 2.1.1.1

Apply the distributive property.

$t = x (x^{2} + x) - 5 (x^{2} + x) + x - 5$

Step 2.1.1.2

Apply the distributive property.

$t = x \cdot x^{2} + x \cdot x - 5 (x^{2} + x) + x - 5$

Step 2.1.1.3

Apply the distributive property.

$t = x \cdot x^{2} + x \cdot x - 5 x^{2} - 5 x + x - 5$

Step 2.1.2

Simplify and combine like terms.

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Step 2.1.2.1

Simplify each term.

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Step 2.1.2.1.1

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

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Step 2.1.2.1.1.1

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

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Step 2.1.2.1.1.1.1

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

$t = x^{1} x^{2} + x \cdot x - 5 x^{2} - 5 x + x - 5$

Step 2.1.2.1.1.1.2

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

$t = x^{1 + 2} + x \cdot x - 5 x^{2} - 5 x + x - 5$

Step 2.1.2.1.1.2

Add $1$ and $2$ .

$t = x^{3} + x \cdot x - 5 x^{2} - 5 x + x - 5$

Step 2.1.2.1.2

Multiply $x$ by $x$ .

$t = x^{3} + x^{2} - 5 x^{2} - 5 x + x - 5$

Step 2.1.2.2

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

$t = x^{3} - 4 x^{2} - 5 x + x - 5$

Step 2.2

Add $- 5 x$ and $x$ .

$t = x^{3} - 4 x^{2} - 4 x - 5$

Step 3

To rewrite as a function of $x$ , write the equation so that $t$ is by itself on one side of the equal sign and an expression involving only $x$ is on the other side.

$f (x) = x^{3} - 4 x^{2} - 4 x - 5$