Algebra Examples

$\frac{k + 3}{4 k - 2} \cdot (12 k^{2} + 2 k - 4)$

Step 1

Factor $2$ out of $4 k - 2$ .

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Step 1.1

Factor $2$ out of $4 k$ .

$\frac{k + 3}{2 (2 k) - 2} \cdot (12 k^{2} + 2 k - 4)$

Step 1.2

Factor $2$ out of $- 2$ .

$\frac{k + 3}{2 (2 k) + 2 (- 1)} \cdot (12 k^{2} + 2 k - 4)$

Step 1.3

Factor $2$ out of $2 (2 k) + 2 (- 1)$ .

$\frac{k + 3}{2 (2 k - 1)} \cdot (12 k^{2} + 2 k - 4)$

Step 2

Multiply $\frac{k + 3}{2 (2 k - 1)}$ by $12 k^{2} + 2 k - 4$ .

$\frac{(k + 3) (12 k^{2} + 2 k - 4)}{2 (2 k - 1)}$

Step 3

Cancel the common factor of $12 k^{2} + 2 k - 4$ and $2$ .

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Step 3.1

Factor $2$ out of $(k + 3) (12 k^{2} + 2 k - 4)$ .

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

Step 3.2

Cancel the common factors.

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 6 \cdot - 2 = - 12$ and whose sum is $b = 1$ .

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Step 4.1.1

Multiply by $1$ .

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

Step 4.1.2

Rewrite $1$ as $- 3$ plus $4$

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

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

Step 4.2.2

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

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

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $2 k - 1$ .

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

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

Step 5

Cancel the common factor of $2 k - 1$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $(k + 3) (3 k + 2)$ by $1$ .

$(k + 3) (3 k + 2)$

Step 6

Expand $(k + 3) (3 k + 2)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$k (3 k + 2) + 3 (3 k + 2)$

Step 6.2

Apply the distributive property.

$k (3 k) + k \cdot 2 + 3 (3 k + 2)$

Step 6.3

Apply the distributive property.

$k (3 k) + k \cdot 2 + 3 (3 k) + 3 \cdot 2$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite using the commutative property of multiplication.

$3 k \cdot k + k \cdot 2 + 3 (3 k) + 3 \cdot 2$

Step 7.1.2

Multiply $k$ by $k$ by adding the exponents.

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Step 7.1.2.1

Move $k$ .

$3 (k \cdot k) + k \cdot 2 + 3 (3 k) + 3 \cdot 2$

Step 7.1.2.2

Multiply $k$ by $k$ .

$3 k^{2} + k \cdot 2 + 3 (3 k) + 3 \cdot 2$

Step 7.1.3

Move $2$ to the left of $k$ .

$3 k^{2} + 2 \cdot k + 3 (3 k) + 3 \cdot 2$

Step 7.1.4

Multiply $3$ by $3$ .

$3 k^{2} + 2 k + 9 k + 3 \cdot 2$

Step 7.1.5

Multiply $3$ by $2$ .

$3 k^{2} + 2 k + 9 k + 6$

Step 7.2

Add $2 k$ and $9 k$ .

$3 k^{2} + 11 k + 6$