UPSKILL MATH PLUS
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Learn moreRelation between zeroes and the coefficients of a linear polynomial:
If \(\alpha\) is the zero of the linear polynomial \(p(x) = ax + b\), then \(\alpha = \frac{-b}{a}\).
This implies \(\alpha\) \(=\) \(\frac{-\left(\text{Constant term}\right)}{\text{Coefficient of }x}\).
Relation between zeroes and the coefficients of a quadratic polynomial:
If \(\alpha\) and \(\beta\) are the zeroes of the quadratic polynomial \(p(x) = ax^2 + bx + c\), then.
(i) \(\alpha + \beta\) \(=\) \(\frac{-b}{a}\)
This implies sum of the zeroes \(=\) \(\frac{-\left(\text{Coefficient of }x\right)}{\text{Coefficient of }x^2}\).
(ii) \(\alpha \beta\) \(=\) \(\frac{c}{a}\)
This implies product of the zeroes \(=\) \(\frac{\text{Constant term}}{\text{Coefficient of }x^2}\).
Important!
The general form of the quadratic polynomial based on its zeroes is given by:
\(p(x)\) \(=\) \(x^2 - \left(\text{Sum of the zeroes}\right)x + \left(\text{Product of the zeroes}\right)\)
Relation between zeroes and the coefficients of a cubic polynomial:
If \(\alpha\), \(\beta\) and \(\gamma\) are the zeroes of the cubic polynomial \(p(x) = ax^3 + bx^2 + cx + d\), then:
(i) \(\alpha + \beta + \gamma\) \(=\) \(\frac{-b}{a}\)
This implies, \(\alpha + \beta + \gamma\) \(=\) \(\frac{-\left(\text{Coefficient of }x^2\right)}{\text{Coefficient of }x^3}\).
(ii) \(\alpha \beta + \beta \gamma + \gamma \alpha\) \(=\) \(\frac{c}{a}\)
This implies, \(\alpha \beta + \beta \gamma + \gamma \alpha\) \(=\) \(\frac{\text{Coefficient of }x}{\text{Coefficient of }x^3}\).
(iii) \(\alpha \beta \gamma\) \(=\) \(\frac{-d}{a}\)
This implies, \(\alpha \beta \gamma\) \(=\) \(\frac{-\left(\text{Constant term}\right)}{\text{Coefficient of }x^3}\).
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