Simplify (45-5w)/(3w^2-28w+9)

$\frac{45 - 5 w}{3 w^{2} - 28 w + 9}$

Step 1

Factor $5$ out of $45 - 5 w$ .

Factor $5$ out of $45$ .

$\frac{5 (9) - 5 w}{3 w^{2} - 28 w + 9}$

Factor $5$ out of $- 5 w$ .

$\frac{5 (9) + 5 (- w)}{3 w^{2} - 28 w + 9}$

Factor $5$ out of $5 (9) + 5 (- w)$ .

$\frac{5 (9 - w)}{3 w^{2} - 28 w + 9}$

Step 2

Factor by grouping.

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 $a \cdot c = 3 \cdot 9 = 27$ and whose sum is $b = - 28$ .

Factor $- 28$ out of $- 28 w$ .

$\frac{5 (9 - w)}{3 w^{2} - 28 (w) + 9}$

Rewrite $- 28$ as $- 1$ plus $- 27$

$\frac{5 (9 - w)}{3 w^{2} + (- 1 - 27) w + 9}$

Apply the distributive property.

$\frac{5 (9 - w)}{3 w^{2} - 1 w - 27 w + 9}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{5 (9 - w)}{(3 w^{2} - 1 w) - 27 w + 9}$

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

$\frac{5 (9 - w)}{w (3 w - 1) - 9 (3 w - 1)}$

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

$\frac{5 (9 - w)}{(3 w - 1) (w - 9)}$

Step 3

Cancel the common factor of $9 - w$ and $w - 9$ .

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{5 (- 1 (- 9) - w)}{(3 w - 1) (w - 9)}$

Factor $- 1$ out of $- w$ .

$\frac{5 (- 1 (- 9) - (w))}{(3 w - 1) (w - 9)}$

Factor $- 1$ out of $- 1 (- 9) - (w)$ .

$\frac{5 (- 1 (- 9 + w))}{(3 w - 1) (w - 9)}$

Reorder terms.

$\frac{5 (- 1 (w - 9))}{(3 w - 1) (w - 9)}$

Cancel the common factor.

$\frac{5 (- 1 (w - 9))}{(3 w - 1) (w - 9)}$

Rewrite the expression.

$\frac{5 \cdot (- 1)}{3 w - 1}$

Step 4

Multiply $5$ by $- 1$ .

$\frac{- 5}{3 w - 1}$

Step 5

Move the negative in front of the fraction.

$- \frac{5}{3 w - 1}$

Simplify (45-5w)/(3w^2-28w+9)

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