Find the values of $ a $ and $ b $, if 2 and 3 are zeroes of $x^3+ax^2+bx-30$.

Given :

The given polynomial is $x^3+ax^2+bx-30$.

2 and 3 are zeroes of $x^3+ax^2+bx-30$.

To do :

We have to find the values of a and b.

Solution :

2 and 3 are zeroes of $x^3+ax^2+bx-30$.

At $x = 2$,

$g(2) = (2)^3 + a(2)^2 + b(2) - 30 = 0$.

$8+4a + 2b - 30 = 0$

$4a + 2b = 22$

$2(2a+b)=2(11)$

$2a+b=11$-----(i)

At $x =3$,

$g(3) =(3)^3 + a(3)^2 + b(3) - 30 = 0$

$27+9a + 3b - 30 = 0$

$9a + 3b = 3$

$3(3a+b)=3$

$3a+b=1$----(ii)

Subtracting (i) from (ii), we get,

$(3a+b)-(2a+b) = 1-11$

$3a-2a = -10$

$a=-10$

Substitute $a = -10$ in equation (i)

$2(-10)+b = 11$

$-20+b =11$

$b = 11+20$

$b = 31$.

The value of $a$ is $-10$ and $b$ is $31$.