If x=2 and x=0 are zeroes of the polynomial 2x3-5x2+ax+b, find the values of a and b.

It is given that

x = 2

and

x = 0

are zeroes of the polynomial

2 x^{3} - 5 x^{2} + ax + b

.
We know that zero of a polynomial p(x) is a number ‘a’, where p(a)=0.
Therefore, equate the given polynomial

2 x^{3} - 5 x^{2} + ax + b

to 0.
⇒

2 x^{3} - 5 x^{2} + ax + b = 0

\dots \dots \dots . (1)

substituting the value of x=2 in equation (1), we get,
⇒

2 {(2)}^{3} - 5 {(2)}^{2} + a (2) + b = 0

⇒

- 4 + 2 a + b = 0

…. eq(ii)
Again, substituting the value of x=0 in equation (1), we get,
⇒

2 {(0)}^{3} - 5 {(0)}^{2} + a (0) + b = 0

⇒

0 - 0 + 0 + b = 0

⇒

b = 0

Put b=0 in equation (2), we get,
⇒

- 4 + 2 a + (0) = 0

⇒

a = 2

Therefore, the values of

a = 2

and

b = 0

.
Hence, option 1 is correct.

If $x = 2$ and $x = 0$ are zeroes of the polynomial $2 x^{3} - 5 x^{2} + ax + b$ , find the values of a and b.