Algebra Examples

$(4 f^{5} - 6 f^{4} + 12 f^{3} - 8 f^{2}) {(4 f^{2})}^{- 1}$

Step 1

To write a polynomial in standard form, simplify and then arrange the terms in descending order.

$a x^{2} + b x + c$

Step 2

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

$(4 f^{5} - 6 f^{4} + 12 f^{3} - 8 f^{2}) \frac{1}{4 f^{2}}$

Step 3

Multiply $4 f^{5} - 6 f^{4} + 12 f^{3} - 8 f^{2}$ by $\frac{1}{4 f^{2}}$ .

$\frac{4 f^{5} - 6 f^{4} + 12 f^{3} - 8 f^{2}}{4 f^{2}}$

Step 4

Factor $2 f^{2}$ out of $4 f^{5} - 6 f^{4} + 12 f^{3} - 8 f^{2}$ .

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Step 4.1

Factor $2 f^{2}$ out of $4 f^{5}$ .

$\frac{2 f^{2} (2 f^{3}) - 6 f^{4} + 12 f^{3} - 8 f^{2}}{4 f^{2}}$

Step 4.2

Factor $2 f^{2}$ out of $- 6 f^{4}$ .

$\frac{2 f^{2} (2 f^{3}) + 2 f^{2} (- 3 f^{2}) + 12 f^{3} - 8 f^{2}}{4 f^{2}}$

Step 4.3

Factor $2 f^{2}$ out of $12 f^{3}$ .

$\frac{2 f^{2} (2 f^{3}) + 2 f^{2} (- 3 f^{2}) + 2 f^{2} (6 f) - 8 f^{2}}{4 f^{2}}$

Step 4.4

Factor $2 f^{2}$ out of $- 8 f^{2}$ .

$\frac{2 f^{2} (2 f^{3}) + 2 f^{2} (- 3 f^{2}) + 2 f^{2} (6 f) + 2 f^{2} (- 4)}{4 f^{2}}$

Step 4.5

Factor $2 f^{2}$ out of $2 f^{2} (2 f^{3}) + 2 f^{2} (- 3 f^{2})$ .

$\frac{2 f^{2} (2 f^{3} - 3 f^{2}) + 2 f^{2} (6 f) + 2 f^{2} (- 4)}{4 f^{2}}$

Step 4.6

Factor $2 f^{2}$ out of $2 f^{2} (2 f^{3} - 3 f^{2}) + 2 f^{2} (6 f)$ .

$\frac{2 f^{2} (2 f^{3} - 3 f^{2} + 6 f) + 2 f^{2} (- 4)}{4 f^{2}}$

Step 4.7

Factor $2 f^{2}$ out of $2 f^{2} (2 f^{3} - 3 f^{2} + 6 f) + 2 f^{2} (- 4)$ .

$\frac{2 f^{2} (2 f^{3} - 3 f^{2} + 6 f - 4)}{4 f^{2}}$

Step 5

Cancel the common factor of $2$ and $4$ .

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Step 5.1

Factor $2$ out of $2 f^{2} (2 f^{3} - 3 f^{2} + 6 f - 4)$ .

$\frac{2 (f^{2} (2 f^{3} - 3 f^{2} + 6 f - 4))}{4 f^{2}}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $4 f^{2}$ .

$\frac{2 (f^{2} (2 f^{3} - 3 f^{2} + 6 f - 4))}{2 (2 f^{2})}$

Step 5.2.2

Cancel the common factor.

$\frac{2 (f^{2} (2 f^{3} - 3 f^{2} + 6 f - 4))}{2 (2 f^{2})}$

Step 5.2.3

Rewrite the expression.

$\frac{f^{2} (2 f^{3} - 3 f^{2} + 6 f - 4)}{2 f^{2}}$

Step 6

Cancel the common factor of $f^{2}$ .

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Step 6.1

Cancel the common factor.

$\frac{f^{2} (2 f^{3} - 3 f^{2} + 6 f - 4)}{2 f^{2}}$

Step 6.2

Rewrite the expression.

$\frac{2 f^{3} - 3 f^{2} + 6 f - 4}{2}$

Step 7

Split the fraction $\frac{2 f^{3} - 3 f^{2} + 6 f - 4}{2}$ into two fractions.

$\frac{2 f^{3} - 3 f^{2} + 6 f}{2} + \frac{- 4}{2}$

Step 8

Simplify each term.

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Step 8.1

Factor $f$ out of $2 f^{3} - 3 f^{2} + 6 f$ .

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Step 8.1.1

Factor $f$ out of $2 f^{3}$ .

$\frac{f (2 f^{2}) - 3 f^{2} + 6 f}{2} + \frac{- 4}{2}$

Step 8.1.2

Factor $f$ out of $- 3 f^{2}$ .

$\frac{f (2 f^{2}) + f (- 3 f) + 6 f}{2} + \frac{- 4}{2}$

Step 8.1.3

Factor $f$ out of $6 f$ .

$\frac{f (2 f^{2}) + f (- 3 f) + f \cdot 6}{2} + \frac{- 4}{2}$

Step 8.1.4

Factor $f$ out of $f (2 f^{2}) + f (- 3 f)$ .

$\frac{f (2 f^{2} - 3 f) + f \cdot 6}{2} + \frac{- 4}{2}$

Step 8.1.5

Factor $f$ out of $f (2 f^{2} - 3 f) + f \cdot 6$ .

$\frac{f (2 f^{2} - 3 f + 6)}{2} + \frac{- 4}{2}$

Step 8.2

Divide $- 4$ by $2$ .

$\frac{f (2 f^{2} - 3 f + 6)}{2} - 2$

Step 9

Reorder terms.

$\frac{1}{2} f (2 f^{2} - 3 f + 6) - 2$