Algebra Examples

$f (x) = 2 x^{2} + x - 3$ and $g (x) = x - 1$

Step 1

Find the quotient of the functions.

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Step 1.1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

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

Step 1.2

Simplify.

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Step 1.2.1

Factor by grouping.

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Step 1.2.1.1

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 $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = 1$ .

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Step 1.2.1.1.1

Multiply by $1$ .

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

Step 1.2.1.1.2

Rewrite $1$ as $- 2$ plus $3$

$\frac{2 x^{2} + (- 2 + 3) x - 3}{x - 1}$

Step 1.2.1.1.3

Apply the distributive property.

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

Step 1.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.1.2.1

Group the first two terms and the last two terms.

$\frac{(2 x^{2} - 2 x) + 3 x - 3}{x - 1}$

Step 1.2.1.2.2

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

$\frac{2 x (x - 1) + 3 (x - 1)}{x - 1}$

Step 1.2.1.3

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

$\frac{(x - 1) (2 x + 3)}{x - 1}$

Step 1.2.2

Cancel the common factor of $x - 1$ .

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Step 1.2.2.1

Cancel the common factor.

$\frac{(x - 1) (2 x + 3)}{x - 1}$

Step 1.2.2.2

Divide $2 x + 3$ by $1$ .

$2 x + 3$

Step 2

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 3