Precalculus Examples

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

Step 1

Factor $x^{\frac{- 1}{2}}$ out of $3 x^{\frac{- 1}{2}} + 4 x^{\frac{1}{2}} + x^{\frac{3}{2}}$ .

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Step 1.1

Factor $x^{\frac{- 1}{2}}$ out of $3 x^{\frac{- 1}{2}}$ .

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

Step 1.2

Factor $x^{\frac{- 1}{2}}$ out of $4 x^{\frac{1}{2}}$ .

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

Step 1.3

Factor $x^{\frac{- 1}{2}}$ out of $x^{\frac{3}{2}}$ .

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

Step 1.4

Factor $x^{\frac{- 1}{2}}$ out of $x^{\frac{- 1}{2}} \cdot 3 + x^{\frac{- 1}{2}} (4 x^{\frac{2}{2}})$ .

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

Step 1.5

Factor $x^{\frac{- 1}{2}}$ out of $x^{\frac{- 1}{2}} \cdot (3 + 4 x^{\frac{2}{2}}) + x^{\frac{- 1}{2}} x^{\frac{4}{2}}$ .

$x^{\frac{- 1}{2}} (3 + 4 x^{\frac{2}{2}} + x^{\frac{4}{2}})$

Step 2

Factor $3 + 4 x^{\frac{2}{2}} + x^{\frac{4}{2}}$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $3$ and whose sum is $4$ .

$1, 3$

Step 2.2

Write the factored form using these integers.

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

Step 3

Divide $2$ by $2$ .

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

Step 4

Simplify.

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

Step 5

Divide $2$ by $2$ .

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

Step 6

Factor.

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Step 6.1

Simplify.

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

Step 6.2

Remove unnecessary parentheses.

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