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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 BYJU'S Exam Prep
Updated on: September 25th, 2023
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.
Table of content
Polynomial
- The words Nominal, which means erms, 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.