Precalculus Examples

$x (x + \frac{1}{5}) (x - 4)$

Step 1

Simplify terms.

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Step 1.1

Apply the distributive property.

$(x \cdot x + x \frac{1}{5}) (x - 4)$

Step 1.2

Multiply $x$ by $x$ .

$(x^{2} + x \frac{1}{5}) (x - 4)$

Step 1.3

Combine $x$ and $\frac{1}{5}$ .

$(x^{2} + \frac{x}{5}) (x - 4)$

Step 2

Expand $(x^{2} + \frac{x}{5}) (x - 4)$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$x^{2} (x - 4) + \frac{x}{5} (x - 4)$

Step 2.2

Apply the distributive property.

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

Step 2.3

Apply the distributive property.

$x^{2} x + x^{2} \cdot - 4 + \frac{x}{5} x + \frac{x}{5} \cdot - 4$

Step 3

Simplify and combine like terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

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

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Step 3.1.1.1

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

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Step 3.1.1.1.1

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

$x^{2} x^{1} + x^{2} \cdot - 4 + \frac{x}{5} x + \frac{x}{5} \cdot - 4$

Step 3.1.1.1.2

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

$x^{2 + 1} + x^{2} \cdot - 4 + \frac{x}{5} x + \frac{x}{5} \cdot - 4$

Step 3.1.1.2

Add $2$ and $1$ .

$x^{3} + x^{2} \cdot - 4 + \frac{x}{5} x + \frac{x}{5} \cdot - 4$

Step 3.1.2

Move $- 4$ to the left of $x^{2}$ .

$x^{3} - 4 \cdot x^{2} + \frac{x}{5} x + \frac{x}{5} \cdot - 4$

Step 3.1.3

Multiply $\frac{x}{5} x$ .

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Step 3.1.3.1

Combine $\frac{x}{5}$ and $x$ .

$x^{3} - 4 x^{2} + \frac{x \cdot x}{5} + \frac{x}{5} \cdot - 4$

Step 3.1.3.2

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

$x^{3} - 4 x^{2} + \frac{x^{1} x}{5} + \frac{x}{5} \cdot - 4$

Step 3.1.3.3

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

$x^{3} - 4 x^{2} + \frac{x^{1} x^{1}}{5} + \frac{x}{5} \cdot - 4$

Step 3.1.3.4

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

$x^{3} - 4 x^{2} + \frac{x^{1 + 1}}{5} + \frac{x}{5} \cdot - 4$

Step 3.1.3.5

Add $1$ and $1$ .

$x^{3} - 4 x^{2} + \frac{x^{2}}{5} + \frac{x}{5} \cdot - 4$

Step 3.1.4

Combine $\frac{x}{5}$ and $- 4$ .

$x^{3} - 4 x^{2} + \frac{x^{2}}{5} + \frac{x \cdot - 4}{5}$

Step 3.1.5

Move $- 4$ to the left of $x$ .

$x^{3} - 4 x^{2} + \frac{x^{2}}{5} + \frac{- 4 \cdot x}{5}$

Step 3.1.6

Move the negative in front of the fraction.

$x^{3} - 4 x^{2} + \frac{x^{2}}{5} - \frac{4 x}{5}$

Step 3.2

To write $- 4 x^{2}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{3} - 4 x^{2} \cdot \frac{5}{5} + \frac{x^{2}}{5} - \frac{4 x}{5}$

Step 3.3

Combine $- 4 x^{2}$ and $\frac{5}{5}$ .

$x^{3} + \frac{- 4 x^{2} \cdot 5}{5} + \frac{x^{2}}{5} - \frac{4 x}{5}$

Step 3.4

Combine the numerators over the common denominator.

$x^{3} + \frac{- 4 x^{2} \cdot 5 + x^{2}}{5} - \frac{4 x}{5}$

Step 3.5

To write $x^{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x^{3} \cdot \frac{5}{5} + \frac{- 4 x^{2} \cdot 5 + x^{2}}{5} - \frac{4 x}{5}$

Step 3.6

Combine $x^{3}$ and $\frac{5}{5}$ .

$\frac{x^{3} \cdot 5}{5} + \frac{- 4 x^{2} \cdot 5 + x^{2}}{5} - \frac{4 x}{5}$

Step 3.7

Combine the numerators over the common denominator.

$\frac{x^{3} \cdot 5 - 4 x^{2} \cdot 5 + x^{2}}{5} - \frac{4 x}{5}$

Step 3.8

Combine the numerators over the common denominator.

$\frac{x^{3} \cdot 5 - 4 x^{2} \cdot 5 + x^{2} - 4 x}{5}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x$ out of $x^{3} \cdot 5 - 4 x^{2} \cdot 5 + x^{2} - 4 x$ .

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Step 4.1.1

Factor $x$ out of $x^{3} \cdot 5$ .

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

Step 4.1.2

Factor $x$ out of $- 4 x^{2} \cdot 5$ .

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

Step 4.1.3

Factor $x$ out of $x^{2}$ .

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

Step 4.1.4

Factor $x$ out of $- 4 x$ .

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

Step 4.1.5

Factor $x$ out of $x (x^{2} \cdot 5) + x (- 4 x \cdot 5)$ .

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

Step 4.1.6

Factor $x$ out of $x (x^{2} \cdot 5 - 4 x \cdot 5) + x \cdot x$ .

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

Step 4.1.7

Factor $x$ out of $x (x^{2} \cdot 5 - 4 x \cdot 5 + x) + x \cdot - 4$ .

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

Step 4.2

Move $5$ to the left of $x^{2}$ .

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

Step 4.3

Multiply $5$ by $- 4$ .

$\frac{x (5 x^{2} - 20 x + x - 4)}{5}$

Step 4.4

Add $- 20 x$ and $x$ .

$\frac{x (5 x^{2} - 19 x - 4)}{5}$

Step 4.5

Factor by grouping.

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Step 4.5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 4 = - 20$ and whose sum is $b = - 19$ .

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Step 4.5.1.1

Factor $- 19$ out of $- 19 x$ .

$\frac{x (5 x^{2} - 19 (x) - 4)}{5}$

Step 4.5.1.2

Rewrite $- 19$ as $1$ plus $- 20$

$\frac{x (5 x^{2} + (1 - 20) x - 4)}{5}$

Step 4.5.1.3

Apply the distributive property.

$\frac{x (5 x^{2} + 1 x - 20 x - 4)}{5}$

Step 4.5.1.4

Multiply $x$ by $1$ .

$\frac{x (5 x^{2} + x - 20 x - 4)}{5}$

Step 4.5.2

Factor out the greatest common factor from each group.

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Step 4.5.2.1

Group the first two terms and the last two terms.

$\frac{x ((5 x^{2} + x) - 20 x - 4)}{5}$

Step 4.5.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (x (5 x + 1) - 4 (5 x + 1))}{5}$

Step 4.5.3

Factor the polynomial by factoring out the greatest common factor, $5 x + 1$ .

$\frac{x ((5 x + 1) (x - 4))}{5}$

$\frac{x (5 x + 1) (x - 4)}{5}$