1) \( \frac{3}{a-4}+\frac{2}{a+3} \cdots \frac{21}{a^{2}-a-12} \)
Solución de inteligencia artificial de Upstudy
Responder
Solución
Respondido por UpStudy AI y revisado por un tutor profesional


Bonus Knowledge
The expression \( \frac{3}{a-4}+\frac{2}{a+3}+\frac{21}{a^{2}-a-12} \) can be simplified by factoring the quadratic in the third term. The quadratic \( a^2 - a - 12 \) factors to \( (a-4)(a+3) \). Thus, we can rewrite the third term as \( \frac{21}{(a-4)(a+3)} \). Now the entire expression is: \[ \frac{3}{a-4} + \frac{2}{a+3} + \frac{21}{(a-4)(a+3)} \] To combine these fractions, we need a common denominator, which is \( (a-4)(a+3) \). Rewriting each term with this common denominator gives: \[ \frac{3(a+3)}{(a-4)(a+3)} + \frac{2(a-4)}{(a-4)(a+3)} + \frac{21}{(a-4)(a+3)} \] Now we can combine the numerators: \[ \frac{3(a+3) + 2(a-4) + 21}{(a-4)(a+3)} \] Distributing in the numerator results in: \[ 3a + 9 + 2a - 8 + 21 = 5a + 22 \] Thus, the simplified expression is: \[ \frac{5a + 22}{(a-4)(a+3)} \] And that's your simplified expression!
