Basic Math Examples

$\frac{15 + 2 y - y^{2}}{3 y^{2} - 16 y + 5}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $15$ as $16$ plus $- 1$

$\frac{16 - 1 + 2 y - y^{2}}{3 y^{2} - 16 y + 5}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{16 - (1^{2} - 2 y + y^{2})}{3 y^{2} - 16 y + 5}$

Step 1.2.2

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

$2 y = 2 \cdot 1 \cdot y$

Step 1.2.3

Rewrite the polynomial.

$\frac{16 - (1^{2} - 2 \cdot 1 \cdot y + y^{2})}{3 y^{2} - 16 y + 5}$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 1$ and $b = y$ .

$\frac{16 - {(1 - y)}^{2}}{3 y^{2} - 16 y + 5}$

Step 1.3

Rewrite $16$ as $4^{2}$ .

$\frac{4^{2} - {(1 - y)}^{2}}{3 y^{2} - 16 y + 5}$

Step 1.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = 1 - y$ .

$\frac{(4 + 1 - y) (4 - (1 - y))}{3 y^{2} - 16 y + 5}$

Step 1.5

Simplify.

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Step 1.5.1

Add $4$ and $1$ .

$\frac{(5 - y) (4 - (1 - y))}{3 y^{2} - 16 y + 5}$

Step 1.5.2

Apply the distributive property.

$\frac{(5 - y) (4 - 1 \cdot 1 - - y)}{3 y^{2} - 16 y + 5}$

Step 1.5.3

Multiply $- 1$ by $1$ .

$\frac{(5 - y) (4 - 1 - - y)}{3 y^{2} - 16 y + 5}$

Step 1.5.4

Multiply $- - y$ .

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Step 1.5.4.1

Multiply $- 1$ by $- 1$ .

$\frac{(5 - y) (4 - 1 + 1 y)}{3 y^{2} - 16 y + 5}$

Step 1.5.4.2

Multiply $y$ by $1$ .

$\frac{(5 - y) (4 - 1 + y)}{3 y^{2} - 16 y + 5}$

Step 1.5.5

Subtract $1$ from $4$ .

$\frac{(5 - y) (3 + y)}{3 y^{2} - 16 y + 5}$

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 $a \cdot c = 3 \cdot 5 = 15$ and whose sum is $b = - 16$ .

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Step 2.1.1

Factor $- 16$ out of $- 16 y$ .

$\frac{(5 - y) (3 + y)}{3 y^{2} - 16 (y) + 5}$

Step 2.1.2

Rewrite $- 16$ as $- 1$ plus $- 15$

$\frac{(5 - y) (3 + y)}{3 y^{2} + (- 1 - 15) y + 5}$

Step 2.1.3

Apply the distributive property.

$\frac{(5 - y) (3 + y)}{3 y^{2} - 1 y - 15 y + 5}$

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.

$\frac{(5 - y) (3 + y)}{(3 y^{2} - 1 y) - 15 y + 5}$

Step 2.2.2

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

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

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $3 y - 1$ .

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

Step 3

Cancel the common factor of $5 - y$ and $y - 5$ .

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Step 3.1

Rewrite $5$ as $- 1 (- 5)$ .

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

Step 3.2

Factor $- 1$ out of $- y$ .

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

Step 3.3

Factor $- 1$ out of $- 1 (- 5) - (y)$ .

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

Step 3.4

Reorder terms.

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

Step 3.5

Cancel the common factor.

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

Step 3.6

Rewrite the expression.

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

Step 4

Move the negative in front of the fraction.

$- \frac{3 + y}{3 y - 1}$