Precalculus Examples

$(x - 2) (x - \frac{1}{2})$

Step 1

Expand $(x - 2) (x - \frac{1}{2})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$x (x - \frac{1}{2}) - 2 (x - \frac{1}{2})$

Step 1.2

Apply the distributive property.

$x \cdot x + x (- \frac{1}{2}) - 2 (x - \frac{1}{2})$

Step 1.3

Apply the distributive property.

$x \cdot x + x (- \frac{1}{2}) - 2 x - 2 (- \frac{1}{2})$

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $x$ by $x$ .

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

Step 2.1.2

Combine $x$ and $\frac{1}{2}$ .

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

Step 2.1.3

Cancel the common factor of $2$ .

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Step 2.1.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

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

Step 2.1.3.2

Factor $2$ out of $- 2$ .

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

Step 2.1.3.3

Cancel the common factor.

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

Step 2.1.3.4

Rewrite the expression.

$x^{2} - \frac{x}{2} - 2 x - 1 \cdot - 1$

Step 2.1.4

Multiply $- 1$ by $- 1$ .

$x^{2} - \frac{x}{2} - 2 x + 1$

Step 2.2

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

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

Step 2.3

Combine $- 2 x$ and $\frac{2}{2}$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 2.5

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

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

Step 2.6

Combine $x^{2}$ and $\frac{2}{2}$ .

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

Step 2.7

Combine the numerators over the common denominator.

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

Step 2.8

Write $1$ as a fraction with a common denominator.

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

Step 2.9

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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Step 3.1

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

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

Step 3.2

Multiply $2$ by $- 2$ .

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

Step 3.3

Subtract $4 x$ from $- x$ .

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

Step 3.4

Factor by grouping.

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Step 3.4.1

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 $a \cdot c = 2 \cdot 2 = 4$ and whose sum is $b = - 5$ .

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Step 3.4.1.1

Factor $- 5$ out of $- 5 x$ .

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

Step 3.4.1.2

Rewrite $- 5$ as $- 1$ plus $- 4$

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

Step 3.4.1.3

Apply the distributive property.

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

Step 3.4.2

Factor out the greatest common factor from each group.

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Step 3.4.2.1

Group the first two terms and the last two terms.

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

Step 3.4.2.2

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

$\frac{x (2 x - 1) - 2 (2 x - 1)}{2}$

Step 3.4.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{(2 x - 1) (x - 2)}{2}$