Question:
If $x=0$ and $x=-1$ are the roots of the polynomial $f(x)=2 x^{3}-3 x^{2}+a x+b$, Find the of $a$ and $b$.
Solution:
We know that, $f(x)=2 x^{3}-3 x^{2}+a x+b$
Given, the values of x are 0 and -1
Substitute x = 0 in f(x)
$f(0)=2(0)^{3}-3(0)^{2}+a(0)+b$
= 0 - 0 + 0 + b
= b .... 1
Substitute x = (-1) in f(x)
$f(-1)=2(-1)^{3}-3(-1)^{2}+a(-1)+b$
= -2 - 3 - a + b
= – 5 - a + b ..... 2
We need to equate equations 1 and 2 to zero
b = 0 and – 5 – a + b = 0
since, the value of b is zero
substitute b = 0 in equation 2
⟹ – 5 – a = – b
⟹ – 5 – a = 0
a = – 5
the values of a and b are - 5 and 0 respectively