Algebra Examples

$\frac{7 r^{5} + 42 r^{4} + 56 r^{3}}{10 r + 5 r^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $7 r^{3}$ out of $7 r^{5} + 42 r^{4} + 56 r^{3}$ .

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Step 1.1.1

Factor $7 r^{3}$ out of $7 r^{5}$ .

$\frac{7 r^{3} (r^{2}) + 42 r^{4} + 56 r^{3}}{10 r + 5 r^{2}}$

Step 1.1.2

Factor $7 r^{3}$ out of $42 r^{4}$ .

$\frac{7 r^{3} (r^{2}) + 7 r^{3} (6 r) + 56 r^{3}}{10 r + 5 r^{2}}$

Step 1.1.3

Factor $7 r^{3}$ out of $56 r^{3}$ .

$\frac{7 r^{3} (r^{2}) + 7 r^{3} (6 r) + 7 r^{3} (8)}{10 r + 5 r^{2}}$

Step 1.1.4

Factor $7 r^{3}$ out of $7 r^{3} (r^{2}) + 7 r^{3} (6 r)$ .

$\frac{7 r^{3} (r^{2} + 6 r) + 7 r^{3} (8)}{10 r + 5 r^{2}}$

Step 1.1.5

Factor $7 r^{3}$ out of $7 r^{3} (r^{2} + 6 r) + 7 r^{3} (8)$ .

$\frac{7 r^{3} (r^{2} + 6 r + 8)}{10 r + 5 r^{2}}$

Step 1.2

Factor $r^{2} + 6 r + 8$ 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 $8$ and whose sum is $6$ .

$2, 4$

Step 1.2.2

Write the factored form using these integers.

$\frac{7 r^{3} ((r + 2) (r + 4))}{10 r + 5 r^{2}}$

$\frac{7 r^{3} (r + 2) (r + 4)}{10 r + 5 r^{2}}$

Step 2

Simplify terms.

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Step 2.1

Factor $5 r$ out of $10 r + 5 r^{2}$ .

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Step 2.1.1

Factor $5 r$ out of $10 r$ .

$\frac{7 r^{3} (r + 2) (r + 4)}{5 r (2) + 5 r^{2}}$

Step 2.1.2

Factor $5 r$ out of $5 r^{2}$ .

$\frac{7 r^{3} (r + 2) (r + 4)}{5 r (2) + 5 r (r)}$

Step 2.1.3

Factor $5 r$ out of $5 r (2) + 5 r (r)$ .

$\frac{7 r^{3} (r + 2) (r + 4)}{5 r (2 + r)}$

Step 2.2

Cancel the common factor of $r^{3}$ and $r$ .

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Step 2.2.1

Factor $r$ out of $7 r^{3} (r + 2) (r + 4)$ .

$\frac{r (7 r^{2} (r + 2) (r + 4))}{5 r (2 + r)}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $r$ out of $5 r (2 + r)$ .

$\frac{r (7 r^{2} (r + 2) (r + 4))}{r (5 (2 + r))}$

Step 2.2.2.2

Cancel the common factor.

$\frac{r (7 r^{2} (r + 2) (r + 4))}{r (5 (2 + r))}$

Step 2.2.2.3

Rewrite the expression.

$\frac{7 r^{2} (r + 2) (r + 4)}{5 (2 + r)}$

Step 2.3

Cancel the common factor of $r + 2$ and $2 + r$ .

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Step 2.3.1

Reorder terms.

$\frac{7 r^{2} (r + 2) (r + 4)}{5 (r + 2)}$

Step 2.3.2

Cancel the common factor.

$\frac{7 r^{2} (r + 2) (r + 4)}{5 (r + 2)}$

Step 2.3.3

Rewrite the expression.

$\frac{7 r^{2} (r + 4)}{5}$