$\left(x - 1\right)$ is a factor of the expression $2x^3 + ax^2 + bx - 14$.
When the expression is divided by $\left(x - 3\right)$, it leaves a remainder $52$.
Find the values of $a$ and $b$.
Let $\;$ $f \left(x\right) = 2x^3 + ax^2 + bx -14$
Given: $\;$ $\left(x - 1\right)$ is a factor of $f \left(x\right)$
Then, by factor theorem, $\;$ $f \left(1\right) = 0$
i.e. $\;$ $f \left(1\right) = 2 \times 1^3 + a \times 1^2 + b \times 1 - 14 = 0$
i.e. $\;$ $a + b = 12$ $\;\;\; \cdots \; (1)$
Given: $\;$ When $f \left(x\right)$ is divided by $\left(x - 3\right)$, it leaves a remainder $52$
Then, by remainder theorem, $\;$ $f \left(3\right) = 52$
i.e. $\;$ $f \left(3\right) = 2 \times 3^3 + a \times 3^2 + b \times 3 - 14 = 52$
i.e. $\;$ $9a + 3b = 12$
i.e. $\;$ $3a + b = 4$ $\;\;\; \cdots \; (2)$
Solving equations $(1)$ and $(2)$ simultaneously, we have
$2a = -8$ $\implies$ $a = -4$
Substituting the value of $a$ in equation $(1)$ gives
$b = 12 - a = 12 - \left(-4\right) = 16$