Question
If $(x-2)$ is a factor of the expression $2 x^3+a x^2+b x-14$ and when the expression is divided by $(x-3)$, it leaves a remainder 52 , find the values of $a$ and $b$.
✓
Answer
Let $f(x)=2 x^3+a x^2+b x-14.....(1)$
as $(x-2)$ is factor of $(1)$
Put $x-2=0$
$\Rightarrow x = 2 in (1)$
$f(2) = 2(2)^3 + a(2)^2 + b(2) - 14$
$0 = 16 + 4a + 2b - 14$
or
$4a + 2b = -2$
$or 2a + b = -1 ...(2)$
Again when f(x) is divided by $(x - 3)$, it leaves remainder $52$
Put $x - 3 = 0$
$\Rightarrow x = 3$
$f(3) = 2(3)^3+ a(3)^2 + b(3) - 14$
$52 = 54 + 9a + 3b - 14$
$52 = 9a + 3b + 40$
$52 - 40 = 9a + 3b$
$\Rightarrow 12 = 9a + 3b$
or
$4 = 3a + b ...(3)$
Solving $(2)$ and $(3)$
$3a + b = 4$
$2a + b = -1$
Sub $- - +$
$a = 5$
Substitute $a = 5$ in $3a + b = 4$
$\Rightarrow 3 x 5 + b = 4$
$15 + b = 4$
$\Rightarrow b = 4 - 15$
$b = -11.$
as $(x-2)$ is factor of $(1)$
Put $x-2=0$
$\Rightarrow x = 2 in (1)$
$f(2) = 2(2)^3 + a(2)^2 + b(2) - 14$
$0 = 16 + 4a + 2b - 14$
or
$4a + 2b = -2$
$or 2a + b = -1 ...(2)$
Again when f(x) is divided by $(x - 3)$, it leaves remainder $52$
Put $x - 3 = 0$
$\Rightarrow x = 3$
$f(3) = 2(3)^3+ a(3)^2 + b(3) - 14$
$52 = 54 + 9a + 3b - 14$
$52 = 9a + 3b + 40$
$52 - 40 = 9a + 3b$
$\Rightarrow 12 = 9a + 3b$
or
$4 = 3a + b ...(3)$
Solving $(2)$ and $(3)$
$3a + b = 4$
$2a + b = -1$
Sub $- - +$
$a = 5$
Substitute $a = 5$ in $3a + b = 4$
$\Rightarrow 3 x 5 + b = 4$
$15 + b = 4$
$\Rightarrow b = 4 - 15$
$b = -11.$
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