Question

If t = 0, t = 2 are the zeroes of 2t^3 - 5t^2 + at + b, then find the values of a and b

Answer 1

Answer:

$\underline{\bigstar{\sf\ Given:-}}$

Polynomial = 2t³ - 5t² + at + b

t = 0 and t = 2 are zeroes of the given polynomial.

$\underline{\bigstar{\sf\ To\:Find:-}}$

Values of a and b

$\underline{\bigstar{\sf\ Solutionn:-}}$

Put t = 0 in the polynomial

2(0)³ - 5(0)² + a(0) + b = 0

⇒ b = 0

Put t = 2 in the given polynomial

2(2)³ - 5(2)² + a(2) + b(2) = 0

16 - 20 + 2a + 2b = 0

- 4 + 2a + 2b = 0

2a + 2b = 4

Put b = 0

2a + 2(0) = 4

2a = 4

a = $\frac {4}{2}$

⇒ a = 2

Answer 2

Solution -

We have an equation,

• p(t) = 2t³ - 5t² + at + b

Zeroes of the given equation are :-

• t = 0
• t = 2

⠀

It means that t = 0 and t = 2 is the solution of the given equation.

⠀

Putting t = 0

⇢ p(0) = 2(0)³ - 5(0)² + a(0) + b

⇢ p(0) = 2(0) - 5(0) + 0 + b

⇢ p(0) = 0 - 0 + 0 + b

⇢ p(0) = b

⠀

Now, if t = 0 is the solution of polynomial given above, then p(0) = 0.

⠀

Put p(0) = 0, we get

• b = 0

⠀

Putting the value of t = 2 and b = 0 in the given polynomial.

⇢ p(2) = 2(2)³ - 5(2)² + a(2) + 0

⇢ p(2) = 2(8) - 5(4) + 2a + 0

⇢ p(2) = 16 - 20 + 2a

⇢ p(2) = 2a - 4

⠀

Again, put p(2) = 0

⇢ 2a - 4 = 0

⇢ 2a = 4

⇢ a = 4/2

⇢ a = 2

⠀

Thus, value of a = 2 and b = 0.