Form new equation

Roots of Polynomial Equations

Basic Concept
Video

Form New Equation

(1) $(x - α) (x - β) (x - γ) = 0$

If $α, β$ and $γ$ are known.

(2) $x^{3} - (α + β + γ) . x^{2} + (α . β + α . γ + β . γ) . x - (α . β . γ) = 0$

If relations of $α, β$ and $γ$ are known.

Example

Find the equation which roots are −4, 3 and 5.

First method

$\begin{aligned} (x - α) (x - β) (x - γ) = 0 \\ (x + 4) (x - 3) (x - 5) = 0 \\ (x^{2} + x - 12) (x - 5) = 0 \\ x^{3} - 4 x^{2} - 17 x + 60 = 0 \end{aligned}$

Second method

$α + β + γ$

$- 4 + 3 + 5$

$4$

$α . β + α . γ + β . γ$

$- 4 . 3 + - 4 . 5 + 3 . 5$

$- 17$

$α . β . γ$

$- 4 . 3 . 5$

$- 60$

Equation

$\begin{aligned} x^{3} - (α + β + γ) x^{2} + (α . β + α . γ + β . γ) x - (α . β . γ) = 0 \\ x^{3} - 4 x^{2} - 17 x + 60 = 0 \end{aligned}$

Exercise

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