Roots of polynomial equations
Roots and coefficients
- For $ax^2 + bx + c = 0$ with roots $\alpha, \beta$:
$$\alpha + \beta = -\frac{b}{a}, \qquad \alpha\beta = \frac{c}{a}$$
- For a cubic $ax^3 + bx^2 + cx + d = 0$: $\sum\alpha = -\dfrac{b}{a}$, $\sum\alpha\beta = \dfrac{c}{a}$, $\alpha\beta\gamma = -\dfrac{d}{a}$.
Practice
For x² − 5x + 6 = 0, what is the sum of the roots α + β = −b/a?
α + β = −b/a = −(−5)/1 = 5.
Practice
For x² − 5x + 6 = 0, what is the product αβ = c/a?
αβ = c/a = 6/1 = 6.
Transforming roots
- To find an equation whose roots are changed simply, use a substitution (e.g. $w = \alpha + 1$).
- Example: $x^2 - 5x + 6 = 0$ ($\alpha+\beta=5$, $\alpha\beta=6$) → roots $\alpha+1,\beta+1$ give sum $7$, product $12$, so $x^2 - 7x + 12 = 0$.
Practice
With α + β = 5 and αβ = 6, what is the product (α+1)(β+1)?
(α+1)(β+1) = αβ + α + β + 1 = 6 + 5 + 1 = 12.
You've got it
Key idea
- quadratic: $\alpha+\beta = -\dfrac{b}{a}$, $\alpha\beta = \dfrac{c}{a}$
- cubic: $\sum\alpha = -\dfrac{b}{a}$, $\sum\alpha\beta = \dfrac{c}{a}$, $\alpha\beta\gamma = -\dfrac{d}{a}$
- transform roots with a substitution