Question 35

Explanation

Factor the first divisor

The divisor is $x^2 + 3x + 2 = (x + 1)(x + 2) \rightarrow x = -1 \vee x = -2$.

The remainder is $s(x) = 3bx + a - 2$.

For $x = -1$

For $x = -2$

Factor the second divisor

For the second polynomial, the divisor is $x^2 - 2x - 3 = (x + 1)(x - 3) \rightarrow x = -1 \vee x = 3$.

The remainder is $s(x) = ax - 2b$.

For $x = -1$

For $x = 3$

Solve the system of equations

From the first and third equations, we get

Then combine equations two, four, and five into

Then the value of $b = 2a - 2 = 2\left(\frac{2}{3}\right) - 2 = -\frac{2}{3}$.

Therefore, $a + b = \frac{2}{3} + \left(-\frac{2}{3}\right) = 0$.