Basic Math Examples

$\frac{\frac{21 y^{2} + 4 y - 1}{5 y + 3}}{\frac{3 y + 1}{10 y^{2} + y - 3}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{21 y^{2} + 4 y - 1}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 2

Factor by grouping.

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Step 2.1

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 $a \cdot c = 21 \cdot - 1 = - 21$ and whose sum is $b = 4$ .

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Step 2.1.1

Factor $4$ out of $4 y$ .

$\frac{21 y^{2} + 4 (y) - 1}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 2.1.2

Rewrite $4$ as $- 3$ plus $7$

$\frac{21 y^{2} + (- 3 + 7) y - 1}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 2.1.3

Apply the distributive property.

$\frac{21 y^{2} - 3 y + 7 y - 1}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(21 y^{2} - 3 y) + 7 y - 1}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 2.2.2

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

$\frac{3 y (7 y - 1) + 1 (7 y - 1)}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $7 y - 1$ .

$\frac{(7 y - 1) (3 y + 1)}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 3

Cancel the common factor of $3 y + 1$ .

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Step 3.1

Factor $3 y + 1$ out of $(7 y - 1) (3 y + 1)$ .

$\frac{(3 y + 1) (7 y - 1)}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 3.2

Cancel the common factor.

$\frac{(3 y + 1) (7 y - 1)}{5 y + 3} \cdot \frac{10 y^{2} + y - 3}{3 y + 1}$

Step 3.3

Rewrite the expression.

$\frac{7 y - 1}{5 y + 3} (10 y^{2} + y - 3)$

Step 4

Multiply $\frac{7 y - 1}{5 y + 3}$ by $10 y^{2} + y - 3$ .

$\frac{(7 y - 1) (10 y^{2} + y - 3)}{5 y + 3}$

Step 5

Factor by grouping.

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Step 5.1

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 $a \cdot c = 10 \cdot - 3 = - 30$ and whose sum is $b = 1$ .

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Step 5.1.1

Multiply by $1$ .

$\frac{(7 y - 1) (10 y^{2} + 1 y - 3)}{5 y + 3}$

Step 5.1.2

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

$\frac{(7 y - 1) (10 y^{2} + (- 5 + 6) y - 3)}{5 y + 3}$

Step 5.1.3

Apply the distributive property.

$\frac{(7 y - 1) (10 y^{2} - 5 y + 6 y - 3)}{5 y + 3}$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$\frac{(7 y - 1) ((10 y^{2} - 5 y) + 6 y - 3)}{5 y + 3}$

Step 5.2.2

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

$\frac{(7 y - 1) (5 y (2 y - 1) + 3 (2 y - 1))}{5 y + 3}$

Step 5.3

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

$\frac{(7 y - 1) ((2 y - 1) (5 y + 3))}{5 y + 3}$

$\frac{(7 y - 1) (2 y - 1) (5 y + 3)}{5 y + 3}$

Step 6

Cancel the common factor of $5 y + 3$ .

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Step 6.1

Cancel the common factor.

$\frac{(7 y - 1) (2 y - 1) (5 y + 3)}{5 y + 3}$

Step 6.2

Divide $(7 y - 1) (2 y - 1)$ by $1$ .

$(7 y - 1) (2 y - 1)$

Step 7

Expand $(7 y - 1) (2 y - 1)$ using the FOIL Method.

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Step 7.1

Apply the distributive property.

$7 y (2 y - 1) - 1 (2 y - 1)$

Step 7.2

Apply the distributive property.

$7 y (2 y) + 7 y \cdot - 1 - 1 (2 y - 1)$

Step 7.3

Apply the distributive property.

$7 y (2 y) + 7 y \cdot - 1 - 1 (2 y) - 1 \cdot - 1$

Step 8

Simplify and combine like terms.

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Step 8.1

Simplify each term.

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Step 8.1.1

Rewrite using the commutative property of multiplication.

$7 \cdot 2 y \cdot y + 7 y \cdot - 1 - 1 (2 y) - 1 \cdot - 1$

Step 8.1.2

Multiply $y$ by $y$ by adding the exponents.

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Step 8.1.2.1

Move $y$ .

$7 \cdot 2 (y \cdot y) + 7 y \cdot - 1 - 1 (2 y) - 1 \cdot - 1$

Step 8.1.2.2

Multiply $y$ by $y$ .

$7 \cdot 2 y^{2} + 7 y \cdot - 1 - 1 (2 y) - 1 \cdot - 1$

Step 8.1.3

Multiply $7$ by $2$ .

$14 y^{2} + 7 y \cdot - 1 - 1 (2 y) - 1 \cdot - 1$

Step 8.1.4

Multiply $- 1$ by $7$ .

$14 y^{2} - 7 y - 1 (2 y) - 1 \cdot - 1$

Step 8.1.5

Multiply $2$ by $- 1$ .

$14 y^{2} - 7 y - 2 y - 1 \cdot - 1$

Step 8.1.6

Multiply $- 1$ by $- 1$ .

$14 y^{2} - 7 y - 2 y + 1$

Step 8.2

Subtract $2 y$ from $- 7 y$ .

$14 y^{2} - 9 y + 1$