Zeros of a Cubic Polynomial
2026-02-21 20:39 Diff
https://brightchamps.com/en-us/math/algebra/zeroes-of-a-cubic-polynomial

Let α, β, and γ be the cubic polynomial \( ax^3 + bx^2 + cx + d = 0 \), where a ≠ 0 and a, b, and c are the coefficients of \(x^3\), \(x^2\), and x, and d is the constant term. Next, a cubic polynomial’s product of zeros is provided as follows:

Product of zeros of a cubic polynomial = \( -\dfrac{\text{constant term}}{\text{coefficient of } x^3} \).

⇒ \( \alpha \beta \gamma = -\frac{d}{a} \).

Example: Determine the cubic polynomial \(\ kx^3 - 5x^2 - 12x + k = 0 \ \)
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If the polynomial is \(\ kx^3 - 5x^2 - 12x + k = 0 \), then we have a = k, and d = k.

Product of zeros = -(constant term)/(coefficient of \(x^3\))

⇒\(\ \alpha \beta \gamma = -\frac{k}{k} = -1 \ \)

Therefore, the product of the zeros of a cubic polynomial is -1.

Sum of Zeros of a Cubic Polynomial
Let α, β, and γ be the zeros of the cubic polynomial \(\ ax^3 + bx^2 + cx + d = 0 \ \) where a ≠ 0, where a, b, and c are the coefficients of \(x^3\),\(x^2\), and x and d is the constant term. Next, a cubic polynomial’s sum of zeros is expressed as follows:

\(\ \text{The sum of zeros of a cubic polynomial} = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} \ \)

⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).

Example: Determine the cubic polynomial \(\ 5x^3 - 15x^2 - 12x + 27 = 0 \ \).

\(\ \text{Sum of zeros of a cubic polynomial} = -\frac{\text{coefficient of } x^2}{\text{coefficient of } x^3} \ \)
⇒\(\ \alpha + \beta + \gamma = -\frac{b}{a} \ \).

If the polynomial is \(\ 5x^3 - 15x^2 - 12x + 27 = 0 \ \), then we have
a= 5 and b = -15.

Sum of zero = -ba

⇒ \(\ \alpha + \beta + \gamma = -\left(-\frac{15}{5}\right) = 3 \ \)

Therefore, the given cubic polynomial’s sum of zeros is 3.