α, β and γ are the zeroes of cubic polynomial p(x) = ax3 + bx2 + cx + d, (a ≠ 0). The product of their zeroes [αβγ] is?

By Ritesh|Updated : November 14th, 2022

α, β and γ are the zeroes of cubic polynomial p(x) = ax3 + bx2 + cx + d, (a ≠ 0). The product of their zeroes [αβγ] is -d/a.

It is given that p(x) = ax3 + bx2 + cx + d, (a ≠ 0) …. (1)

If α, β, and γ are the zeroes of the cubic polynomial P(x).

Then, we can write as:

p(x) = (x - α) (x - β) (x - γ)

The above equation becomes

p(x) = x3 - (α + β + γ)x2 + (αβ + βγ + γα)x - αβγ …. (2)

Due to the similarity of equations (1) and (2).

The result of comparing the coefficients is:

a/1 = b/ -α - β - γ = c/αβ + βγ + γα = d/-αβγ

On solving, the above equation we get:

α + β + γ = -b/a = sum of roots

αβ + βγ + γα = c/a = sum of product of the roots

αβγ = -d/a = product of the roots

Hence, the product of zeroes is αβγ = -d/a.

Polynomial

  • The words Nominal, which means "terms," and Poly, which means "many," are combined to make the word polynomial.
  • A polynomial is an equation made up of exponents, constants, and variables that are combined using mathematical operations including addition, subtraction, multiplication, and division (No division operation by a variable).
  • Depending on how many terms are present, the expression is classified as a monomial, binomial, or trinomial.

Summary:

α, β and γ are the zeroes of cubic polynomial p(x) = ax3 + bx2 + cx + d, (a ≠ 0). The product of their zeroes [αβγ] is?

α, β and γ are the zeroes of cubic polynomial p(x) = ax3 + bx2 + cx + d, (a ≠ 0). The product of their zeroes [αβγ] is -d/a. P(x) stands for the polynomial function, where x stands for the variable.

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