Question

and correct the error in factoring the polynomial. 24. \\( s^2 - 17s - 60 = (s - 5)(s - 12) \\)

Explanation:

Step1: Identify error in given factors

Expand $(s-5)(s-12)$:

$$\begin{align*} (s-5)(s-12)&=s^2-12s-5s+60\\ &=s^2-17s+60 \end{align*}$$

The constant term is $+60$, but original polynomial has $-60$.

Step2: Find correct factor pair

Find two numbers that multiply to $-60$ and add to $-17$. The pair is $-20$ and $+3$, since $(-20)\times3=-60$ and $-20+3=-17$.

Step3: Write correct factored form

Use the valid factor pair to factor the polynomial:
$s^2-17s-60=(s-20)(s+3)$

Answer:

The error is that the product of $(s-5)(s-12)$ gives a constant term of $+60$ instead of $-60$. The correct factorization is $\boldsymbol{(s-20)(s+3)}$.