Algebra Examples

$\frac{x^{6} - 4 x^{4} + 3 x^{2}}{5 x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $x^{2}$ out of $x^{6} - 4 x^{4} + 3 x^{2}$ .

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Step 1.1.1

Factor $x^{2}$ out of $x^{6}$ .

$\frac{x^{2} x^{4} - 4 x^{4} + 3 x^{2}}{5 x^{2}}$

Step 1.1.2

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

$\frac{x^{2} x^{4} + x^{2} (- 4 x^{2}) + 3 x^{2}}{5 x^{2}}$

Step 1.1.3

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

$\frac{x^{2} x^{4} + x^{2} (- 4 x^{2}) + x^{2} \cdot 3}{5 x^{2}}$

Step 1.1.4

Factor $x^{2}$ out of $x^{2} x^{4} + x^{2} (- 4 x^{2})$ .

$\frac{x^{2} (x^{4} - 4 x^{2}) + x^{2} \cdot 3}{5 x^{2}}$

Step 1.1.5

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

$\frac{x^{2} (x^{4} - 4 x^{2} + 3)}{5 x^{2}}$

Step 1.2

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$\frac{x^{2} ({(x^{2})}^{2} - 4 x^{2} + 3)}{5 x^{2}}$

Step 1.3

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$\frac{x^{2} (u^{2} - 4 u + 3)}{5 x^{2}}$

Step 1.4

Factor $u^{2} - 4 u + 3$ using the AC method.

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Step 1.4.1

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 $- 4$ .

$- 3, - 1$

Step 1.4.2

Write the factored form using these integers.

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

Step 1.5

Replace all occurrences of $u$ with $x^{2}$ .

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

Step 1.6

Rewrite $1$ as $1^{2}$ .

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

Step 1.7

Factor.

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

Step 2

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

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

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

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

Step 3

Expand $(x^{2} - 3) (x + 1)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Apply the distributive property.

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

Step 4

Simplify each term.

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Step 4.1

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 4.1.1

Multiply $x^{2}$ by $x$ .

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Step 4.1.1.1

Raise $x$ to the power of $1$ .

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

Step 4.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.1.2

Add $2$ and $1$ .

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

Step 4.2

Multiply $x^{2}$ by $1$ .

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

Step 4.3

Multiply $- 3$ by $1$ .

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

Step 5

Expand $(x^{3} + x^{2} - 3 x - 3) (x - 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 6

Simplify each term.

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Step 6.1

Multiply $x^{3}$ by $x$ by adding the exponents.

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Step 6.1.1

Multiply $x^{3}$ by $x$ .

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Step 6.1.1.1

Raise $x$ to the power of $1$ .

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

Step 6.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 6.1.2

Add $3$ and $1$ .

$\frac{x^{4} + x^{3} \cdot - 1 + x^{2} x + x^{2} \cdot - 1 - 3 x \cdot x - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.2

Move $- 1$ to the left of $x^{3}$ .

$\frac{x^{4} - 1 \cdot x^{3} + x^{2} x + x^{2} \cdot - 1 - 3 x \cdot x - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.3

Rewrite $- 1 x^{3}$ as $- x^{3}$ .

$\frac{x^{4} - x^{3} + x^{2} x + x^{2} \cdot - 1 - 3 x \cdot x - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.4

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 6.4.1

Multiply $x^{2}$ by $x$ .

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Step 6.4.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{4} - x^{3} + x^{2} x^{1} + x^{2} \cdot - 1 - 3 x \cdot x - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x^{4} - x^{3} + x^{2 + 1} + x^{2} \cdot - 1 - 3 x \cdot x - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.4.2

Add $2$ and $1$ .

$\frac{x^{4} - x^{3} + x^{3} + x^{2} \cdot - 1 - 3 x \cdot x - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.5

Move $- 1$ to the left of $x^{2}$ .

$\frac{x^{4} - x^{3} + x^{3} - 1 \cdot x^{2} - 3 x \cdot x - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.6

Rewrite $- 1 x^{2}$ as $- x^{2}$ .

$\frac{x^{4} - x^{3} + x^{3} - x^{2} - 3 x \cdot x - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.7

Multiply $x$ by $x$ by adding the exponents.

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Step 6.7.1

Move $x$ .

$\frac{x^{4} - x^{3} + x^{3} - x^{2} - 3 (x \cdot x) - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.7.2

Multiply $x$ by $x$ .

$\frac{x^{4} - x^{3} + x^{3} - x^{2} - 3 x^{2} - 3 x \cdot - 1 - 3 x - 3 \cdot - 1}{5}$

Step 6.8

Multiply $- 1$ by $- 3$ .

$\frac{x^{4} - x^{3} + x^{3} - x^{2} - 3 x^{2} + 3 x - 3 x - 3 \cdot - 1}{5}$

Step 6.9

Multiply $- 3$ by $- 1$ .

$\frac{x^{4} - x^{3} + x^{3} - x^{2} - 3 x^{2} + 3 x - 3 x + 3}{5}$

Step 7

Combine the opposite terms in $x^{4} - x^{3} + x^{3} - x^{2} - 3 x^{2} + 3 x - 3 x + 3$ .

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Step 7.1

Add $- x^{3}$ and $x^{3}$ .

$\frac{x^{4} + 0 - x^{2} - 3 x^{2} + 3 x - 3 x + 3}{5}$

Step 7.2

Add $x^{4}$ and $0$ .

$\frac{x^{4} - x^{2} - 3 x^{2} + 3 x - 3 x + 3}{5}$

Step 7.3

Subtract $3 x$ from $3 x$ .

$\frac{x^{4} - x^{2} - 3 x^{2} + 0 + 3}{5}$

Step 7.4

Add $x^{4} - x^{2} - 3 x^{2}$ and $0$ .

$\frac{x^{4} - x^{2} - 3 x^{2} + 3}{5}$

Step 8

Subtract $3 x^{2}$ from $- x^{2}$ .

$\frac{x^{4} - 4 x^{2} + 3}{5}$

Step 9

Split the fraction $\frac{x^{4} - 4 x^{2} + 3}{5}$ into two fractions.

$\frac{x^{4} - 4 x^{2}}{5} + \frac{3}{5}$

Step 10

Split the fraction $\frac{x^{4} - 4 x^{2}}{5}$ into two fractions.

$\frac{x^{4}}{5} + \frac{- 4 x^{2}}{5} + \frac{3}{5}$

Step 11

Move the negative in front of the fraction.

$\frac{x^{4}}{5} - \frac{4 x^{2}}{5} + \frac{3}{5}$