Roots of Polynomial Equations, Rational Functions, and Summation of Series

Practice flashcards covering the relationships between roots and coefficients of polynomials, sketching rational functions with asymptotes, and standard summation formulae for series.

For a quadratic equation $ax^2 + bx + c = 0$, what is the relation between the sum of the roots $\alpha + \beta$ and the coefficients?

$$\alpha + \beta = -\frac{b}{a}$$

What is the product of the roots $\alpha\beta$ for the quadratic equation $ax^2 + bx + c = 0$?

$$\alpha\beta = \frac{c}{a}$$

How is the sum of the squares of the roots $\Sigma \alpha^2$ expressed in terms of $\Sigma \alpha$ and $\Sigma \alpha\beta$?

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

For a cubic equation $ax^3 + bx^2 + cx + d = 0$, what is the sum of product of the roots taken two at a time ($\Sigma \alpha\beta$)?

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

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

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

How is the term $S_n$ defined in the context of polynomial roots?

$$S_n = \alpha^n + \beta^n + \gamma^n + \dots$$

What is the general recurrence formula for a cubic equation $ax^3 + bx^2 + cx + d = 0$?

$$aS_{n+3} + bS_{n+2} + cS_{n+1} + dS_n = 0$$

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

$$\frac{e}{a}$$

What is the value of $S_{-1}$ for any polynomial equation?

It is equal to the negative of the coefficient of the linear term divided by the constant term.

What is the definition of an asymptote?

A line that a curve approaches but does not touch (the curve converges to it).

For a rational function $y = \frac{x+1}{x-1}$, how is the vertical asymptote determined?

By setting the denominator to zero, which gives $x = 1$.

When does an oblique asymptote occur in a rational function?

When the degree of the polynomial in the numerator is exactly one greater than the degree of the denominator.

How is the oblique asymptote found for a function of the form $y = \frac{ax^2 + bx + c}{dx + e}$?

By splitting the function into partial fractions/long division to reach the form $y = Ax + B + \frac{C}{dx+e}$, where $y = Ax + B$ is the oblique asymptote.

What effect does the transformation $y = |f(x)|$ have on a graph?

All negative parts of the original curve (where $y < 0$) are reflected in the $x$-axis.

What is a key feature of the graph of $y = f(|x|)$?

The curve is symmetrical about the $y$-axis, consisting of the original curve for $x > 0$ reflected in the $y$-axis.

What is the standard result for the sum of the first $n$ natural numbers ($\sum_{r=1}^n r$)?

$$\frac{1}{2}n(n + 1)$$

What is the standard result for the sum of the squares of the first $n$ natural numbers ($\sum_{r=1}^n r^2$)?

$$\frac{1}{6}n(n + 1)(2n + 1)$$

What is the standard result for the sum of the cubes of the first $n$ natural numbers ($\sum_{r=1}^n r^3$)?

$$\frac{1}{4}n^2(n + 1)^2$$

What is the 'method of differences' used for in summation of series?

Using partial fractions to express terms as differences so that intermediate terms cancel out when summed.

How is the sum to infinity calculated for a convergent series of the form $S_n = b + \frac{C}{dn+e}$?

By allowing $n \rightarrow \infty$, which makes the fractional term approach zero, leaving the sum as $b$.