Factor 15x^(-1/2)+8x^(1/2)+x^(3/2)

$15 x^{- \frac{1}{2}} + 8 x^{\frac{1}{2}} + x^{\frac{3}{2}}$

Step 1

Factor by grouping.

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 $a \cdot c = 15 \cdot 1 = 15$ and whose sum is $b = 8$ .

Factor $8$ out of $8 x^{\frac{1}{2}}$ .

$15 x^{- \frac{1}{2}} + 8 (x^{\frac{1}{2}}) + x^{\frac{3}{2}}$

Rewrite $8$ as $3$ plus $5$

$15 x^{- \frac{1}{2}} + (3 + 5) x^{\frac{1}{2}} + x^{\frac{3}{2}}$

Apply the distributive property.

$15 x^{- \frac{1}{2}} + 3 x^{\frac{1}{2}} + 5 x^{\frac{1}{2}} + x^{\frac{3}{2}}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(15 x^{- \frac{1}{2}} + 3 x^{\frac{1}{2}}) + 5 x^{\frac{1}{2}} + x^{\frac{3}{2}}$

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

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

Factor the polynomial by factoring out the greatest common factor, $5 + x^{\frac{2}{2}}$ .

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

Step 2

Divide $2$ by $2$ .

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

Step 3

Simplify.

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

Step 4

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(5 + x) (\frac{1}{x^{\frac{1}{2}}} \cdot 3 + x^{\frac{1}{2}})$

Step 5

Combine $\frac{1}{x^{\frac{1}{2}}}$ and $3$ .

$(5 + x) (\frac{3}{x^{\frac{1}{2}}} + x^{\frac{1}{2}})$

Factor 15x^(-1/2)+8x^(1/2)+x^(3/2)

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