α, β and γ are the Zeroes of Cubic Polynomial P(x) = ax^3 + bx^2 + cx + d, (a≠0). Then Product of their Zeroes [αβγ] is?
By BYJU'S Exam Prep
Updated on: October 17th, 2023
α, β and γ are the zeroes of cubic polynomial P(x) = ax3 + bx2 + cx + d, (a≠0). Then product of their zeroes [αβγ] is?
To find the product of the zeroes [αβγ] of the cubic polynomial P(x) = ax3 + bx2 + cx + d, where a ≠ 0, we can follow these steps:
Step 1: Recall Vieta’s formulas for a cubic polynomial:Let α, β, and γ be the zeroes of the cubic polynomial P(x). Vieta’s formulas state the following relationships:
- α + β + γ = -b/a (sum of zeroes)
- αβ + βγ + γα = c/a (sum of pairwise products of zeroes)
- αβγ = -d/a (product of zeroes)
Step 2: Since we need to find the product αβγ, we can directly use the value from Vieta’s formulas.
Table of content
α, β and γ are the Zeroes of Cubic Polynomial P(x) = ax3 + bx2 + cx + d, (a≠0). Then Product of their Zeroes [αβγ] is?
Solution:
To find the product of the zeroes of a cubic polynomial, we can use Vieta’s formulas. Let’s denote the cubic polynomial as P(x) = ax3 + bx2 + cx + d.
According to Vieta’s formulas, the product of the zeroes (zeros) of a cubic polynomial is given by the ratio of the constant term to the coefficient of the highest power of x, with appropriate signs.
In this case, the zeroes are denoted as alpha, beta, and gamma. We know that alpha, beta, and gamma are not equal to 0. Therefore, we can write the product of the zeroes as:
Product of zeroes = (-1)n * d / a
where n is the number of terms in the polynomial. In this case, n = 3.
So, the product of the zeroes is:
Product of zeroes = (-1)3 * d / a = -d/a
Hence, the product of the zeroes of the cubic polynomial P(x) = ax3 + bx2 + cx + d, where alpha, beta, and gamma are the zeroes and not equal to 0, is -d/a.
Answer:
α, β and γ are the Zeroes of Cubic Polynomial P(x) = ax3 + bx2 + cx + d, (a≠0). Then Product of their Zeroes [αβγ] is -d/a
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