Pre-Algebra Examples

$\frac{5 x^{4} - 20 x^{3} + 15 x^{2}}{5 x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $5 x^{2}$ out of $5 x^{4} - 20 x^{3} + 15 x^{2}$ .

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Step 1.1.1

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

$\frac{5 x^{2} (x^{2}) - 20 x^{3} + 15 x^{2}}{5 x^{2}}$

Step 1.1.2

Factor $5 x^{2}$ out of $- 20 x^{3}$ .

$\frac{5 x^{2} (x^{2}) + 5 x^{2} (- 4 x) + 15 x^{2}}{5 x^{2}}$

Step 1.1.3

Factor $5 x^{2}$ out of $15 x^{2}$ .

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

Step 1.1.4

Factor $5 x^{2}$ out of $5 x^{2} (x^{2}) + 5 x^{2} (- 4 x)$ .

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

Step 1.1.5

Factor $5 x^{2}$ out of $5 x^{2} (x^{2} - 4 x) + 5 x^{2} (3)$ .

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

Step 1.2

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

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Step 1.2.1

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 $- 4$ .

$- 3, - 1$

Step 1.2.2

Write the factored form using these integers.

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

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

Step 2

Cancel the common factor of $5$ .

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Step 2.1

Cancel the common factor.

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

Step 2.2

Rewrite the expression.

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $(x - 3) (x - 1)$ by $1$ .

$(x - 3) (x - 1)$

Step 4

Expand $(x - 3) (x - 1)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

$x (x - 1) - 3 (x - 1)$

Step 4.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 3 (x - 1)$

Step 4.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 3 x - 3 \cdot - 1$

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $x$ by $x$ .

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

Step 5.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 3 x - 3 \cdot - 1$

Step 5.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 3 x - 3 \cdot - 1$

Step 5.1.4

Multiply $- 3$ by $- 1$ .

$x^{2} - x - 3 x + 3$

Step 5.2

Subtract $3 x$ from $- x$ .

$x^{2} - 4 x + 3$