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If $2x + 3$ and $x + 2$ are the factors of the polynomial $g(x) = 2x^3 + ax^2 + 27x + b$, then find the values of the constants a and b in the polynomial g(x).
Given :
The given polynomial is $g(x) = 2x^3 + ax^2 + 27x + b$.
$2x + 3$ and $x + 2$ are the factors of the polynomial $g(x) = 2x^3 + ax^2 + 27x + b$.
To do :
We have to find the value of a and b in the polynomial g(x).
Solution :
$2x + 3$ and $x + 2$ are the factors of the polynomial $g(x) = 2x^3 + ax^2 + 27x + b$.
At $x = -2$, $g(-2) = 2(-2)^3 + a(-2)^2 + 27(-2) + b = 0$.
$4a + b - 70 = 0$
$4a + b = 70$-----(1)
At $x = \frac{-3}{2}$, $g(\frac{-3}{2}) = 2(-\frac{-3}{2})^3 + a(-\frac{-3}{2})^2 + 27(-\frac{-3}{2}) + b = 0$.
$9a + 4b - 189 = 0$
$9a + 4b = 189$----(2)
To solve the above two equations we multiply equation (1) by 4 so that 4b gets cancelled and we can find the value of a first.
$4(4a+b) = 4(70)$
$16a+4b = 280$ -----(3)
Now,
Equation (3) $-$ equation (1) is,
$16a+4b = 280$
$-(9a+4b = 189)$
--------------------
$7a = 91$
$a = \frac{91}{7}$
$a = 13$
Substitute $a = 13$ in equation (1)
$4(13)+b = 70$
$52+b =70$
$b = 70-52$
$b = 18$.
The value of a is 13 and b is 18.
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