Roots of polynomial equations flashcards

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For a general polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_0 = 0$, what is the formula for the sum of the roots, $\Sigma \alpha$?

$\Sigma \alpha = -\frac{a_{n-1}}{a_n}$

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For a general polynomial of degree $n$, what is the formula for the product of the roots?

Product $= (-1)^n \frac{a_0}{a_n}$

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For a cubic equation $ax^3+bx^2+cx+d=0$, what is $\Sigma \alpha\beta$?

$\Sigma \alpha\beta = \alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$

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For a cubic equation $ax^3+bx^2+cx+d=0$, what is the product of the roots $\alpha\beta\gamma$?

$\alpha\beta\gamma = -\frac{d}{a}$

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For a quartic equation $ax^4+bx^3+cx^2+dx+e=0$, what is $\Sigma \alpha\beta\gamma$?

$\Sigma \alpha\beta\gamma = \alpha\beta\gamma + \alpha\beta\delta + \alpha\gamma\delta + \beta\gamma\delta = -\frac{d}{a}$

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What is the key algebraic identity used to find the sum of the squares of the roots, $\Sigma \alpha^2$?

$\Sigma \alpha^2 = (\Sigma \alpha)^2 - 2(\Sigma \alpha\beta)$

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How can you express the sum of the reciprocals of the roots, $\Sigma \frac{1}{\alpha}$, in terms of other symmetric sums for a cubic?

By creating a common denominator: $\Sigma \frac{1}{\alpha} = \frac{\beta\gamma + \alpha\gamma + \alpha\beta}{\alpha\beta\gamma} = \frac{\Sigma \alpha\beta}{\alpha\beta\gamma}$

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An equation has roots $\alpha, \beta, \gamma$. How do you find the equation with roots $2\alpha, 2\beta, 2\gamma$?

Let the new root be $y=2x$. Then $x = y/2$. Substitute $x=y/2$ into the original equation and simplify.

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An equation has roots $\alpha, \beta, \gamma$. How do you find the equation with roots $\alpha-1, \beta-1, \gamma-1$?

Let the new root be $y=x-1$. Then $x = y+1$. Substitute $x=y+1$ into the original equation and simplify.

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What is a common mistake when applying Vieta's formulas to an equation like $x^3 - 5x + 4 = 0$?

Forgetting that the coefficient of the $x^2$ term is zero. Here, $a=1, b=0, c=-5, d=4$. This means $\Sigma \alpha = -b/a = 0$.

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What is the pattern of signs for Vieta's formulas?

The signs alternate, starting with a negative for the sum of the roots: $-b/a, +c/a, -d/a, +e/a, \dots$

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What must you always remember to divide by when using Vieta's formulas?

The leading coefficient, $a$ (or $a_n$). If the polynomial is monic (leading coefficient is 1), you can omit this, but it's good practice to always consider it.