Simplify (t^2)/(t-3)+9/(3-t)

$\frac{t^{2}}{t - 3} + \frac{9}{3 - t}$

Step 1

Simplify terms.

Rewrite $3$ as $- 1 (- 3)$ .

$\frac{t^{2}}{t - 3} + \frac{9}{- 1 (- 3) - t}$

Factor $- 1$ out of $- t$ .

$\frac{t^{2}}{t - 3} + \frac{9}{- 1 (- 3) - (t)}$

Factor $- 1$ out of $- 1 (- 3) - (t)$ .

$\frac{t^{2}}{t - 3} + \frac{9}{- 1 (- 3 + t)}$

Simplify the expression.

Move a negative from the denominator of $\frac{9}{- 1 (- 3 + t)}$ to the numerator.

$\frac{t^{2}}{t - 3} + \frac{- 1 \cdot 9}{- 3 + t}$

Reorder terms.

$\frac{t^{2}}{t - 3} + \frac{- 1 \cdot 9}{t - 3}$

Combine the numerators over the common denominator.

$\frac{t^{2} - 1 \cdot 9}{t - 3}$

Step 2

Simplify the numerator.

Rewrite $9$ as $3^{2}$ .

$\frac{t^{2} - 3^{2}}{t - 3}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t$ and $b = 3$ .

$\frac{(t + 3) (t - 3)}{t - 3}$

Step 3

Cancel the common factor of $t - 3$ .

Cancel the common factor.

$\frac{(t + 3) (t - 3)}{t - 3}$

Divide $t + 3$ by $1$ .

$t + 3$