Basic Math Examples

$\frac{3 c^{2} - 5 c - 2}{6 c^{2}} \cdot \frac{4 c^{2} - 8 c}{c^{2} - 4 c + 4}$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 3 \cdot - 2 = - 6$ and whose sum is $b = - 5$ .

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Step 1.1.1

Factor $- 5$ out of $- 5 c$ .

$\frac{3 c^{2} - 5 (c) - 2}{6 c^{2}} \cdot \frac{4 c^{2} - 8 c}{c^{2} - 4 c + 4}$

Step 1.1.2

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{3 c^{2} + (1 - 6) c - 2}{6 c^{2}} \cdot \frac{4 c^{2} - 8 c}{c^{2} - 4 c + 4}$

Step 1.1.3

Apply the distributive property.

$\frac{3 c^{2} + 1 c - 6 c - 2}{6 c^{2}} \cdot \frac{4 c^{2} - 8 c}{c^{2} - 4 c + 4}$

Step 1.1.4

Multiply $c$ by $1$ .

$\frac{3 c^{2} + c - 6 c - 2}{6 c^{2}} \cdot \frac{4 c^{2} - 8 c}{c^{2} - 4 c + 4}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2

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

$\frac{c (3 c + 1) - 2 (3 c + 1)}{6 c^{2}} \cdot \frac{4 c^{2} - 8 c}{c^{2} - 4 c + 4}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 c + 1$ .

$\frac{(3 c + 1) (c - 2)}{6 c^{2}} \cdot \frac{4 c^{2} - 8 c}{c^{2} - 4 c + 4}$

Step 2

Factor $4 c$ out of $4 c^{2} - 8 c$ .

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Step 2.1

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

$\frac{(3 c + 1) (c - 2)}{6 c^{2}} \cdot \frac{4 c (c) - 8 c}{c^{2} - 4 c + 4}$

Step 2.2

Factor $4 c$ out of $- 8 c$ .

$\frac{(3 c + 1) (c - 2)}{6 c^{2}} \cdot \frac{4 c (c) + 4 c (- 2)}{c^{2} - 4 c + 4}$

Step 2.3

Factor $4 c$ out of $4 c (c) + 4 c (- 2)$ .

$\frac{(3 c + 1) (c - 2)}{6 c^{2}} \cdot \frac{4 c (c - 2)}{c^{2} - 4 c + 4}$

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $4$ as $2^{2}$ .

$\frac{(3 c + 1) (c - 2)}{6 c^{2}} \cdot \frac{4 c (c - 2)}{c^{2} - 4 c + 2^{2}}$

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 c = 2 \cdot c \cdot 2$

Step 3.3

Rewrite the polynomial.

$\frac{(3 c + 1) (c - 2)}{6 c^{2}} \cdot \frac{4 c (c - 2)}{c^{2} - 2 \cdot c \cdot 2 + 2^{2}}$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = c$ and $b = 2$ .

$\frac{(3 c + 1) (c - 2)}{6 c^{2}} \cdot \frac{4 c (c - 2)}{{(c - 2)}^{2}}$

Step 4

Simplify terms.

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Step 4.1

Combine.

$\frac{(3 c + 1) (c - 2) (4 c (c - 2))}{6 c^{2} {(c - 2)}^{2}}$

Step 4.2

Cancel the common factor of $c - 2$ and ${(c - 2)}^{2}$ .

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Step 4.2.1

Factor $c - 2$ out of $(3 c + 1) (c - 2) (4 c (c - 2))$ .

$\frac{(c - 2) ((3 c + 1) (4 c (c - 2)))}{6 c^{2} {(c - 2)}^{2}}$

Step 4.2.2

Cancel the common factors.

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Step 4.2.2.1

Factor $c - 2$ out of $6 c^{2} {(c - 2)}^{2}$ .

$\frac{(c - 2) ((3 c + 1) (4 c (c - 2)))}{(c - 2) (6 c^{2} (c - 2))}$

Step 4.2.2.2

Cancel the common factor.

$\frac{(c - 2) ((3 c + 1) (4 c (c - 2)))}{(c - 2) (6 c^{2} (c - 2))}$

Step 4.2.2.3

Rewrite the expression.

$\frac{(3 c + 1) (4 c (c - 2))}{6 c^{2} (c - 2)}$

Step 4.3

Cancel the common factor of $4$ and $6$ .

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Step 4.3.1

Factor $2$ out of $(3 c + 1) (4 c (c - 2))$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $2$ out of $6 c^{2} (c - 2)$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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

Step 4.4

Cancel the common factor of $c$ and $c^{2}$ .

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Step 4.4.1

Factor $c$ out of $(3 c + 1) (2 c (c - 2))$ .

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

Step 4.4.2

Cancel the common factors.

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Step 4.4.2.1

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

$\frac{c ((3 c + 1) (2 (c - 2)))}{c (3 c (c - 2))}$

Step 4.4.2.2

Cancel the common factor.

$\frac{c ((3 c + 1) (2 (c - 2)))}{c (3 c (c - 2))}$

Step 4.4.2.3

Rewrite the expression.

$\frac{(3 c + 1) (2 (c - 2))}{3 c (c - 2)}$

Step 4.5

Cancel the common factor of $c - 2$ .

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Step 4.5.1

Cancel the common factor.

$\frac{(3 c + 1) (2 (c - 2))}{3 c (c - 2)}$

Step 4.5.2

Rewrite the expression.

$\frac{(3 c + 1) \cdot (2)}{3 c}$

Step 4.6

Move $2$ to the left of $3 c + 1$ .

$\frac{2 (3 c + 1)}{3 c}$