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

2 thoughts on “If t = 0, t = 2 are the zeroes of 2t^3 – 5t^2 + at + b, then find the values of a and b”

1. Answer:

   [tex]\underline{\bigstar{\sf\ Given:-}}[/tex]

   Polynomial = 2t³ – 5t² + at + b

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

   [tex]\underline{\bigstar{\sf\ To\:Find:-}}[/tex]

   Values of a and b

   [tex]\underline{\bigstar{\sf\ Solutionn:-}}[/tex]

   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 = [tex] \frac {4}{2} [/tex]

   ⇒ a = 2

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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.

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