Lesson 2.2: Roots of Polynomial Equations

Introduction

In this lesson, we will delve into polynomial equations and their roots, essential concepts that form the backbone of algebra and functions. Understanding how to find roots and their relationship to coefficients will enhance your problem-solving skills and prepare you for future mathematical topics.

Learning Objectives:

By the end of this lesson, students should be able to:

Understand relationships between the roots and coefficients of quadratics, cubics, and quartics (sums and products of roots).
Form a new equation whose roots are a function of the original roots.
Use symmetric functions of roots ($\Sigma \alpha$, $\Sigma \alpha \beta$, $\Sigma \alpha^2$).
State and use the root–coefficient relationships for cubics and quartics.
Evaluate symmetric functions such as $\Sigma \alpha^2$ without finding the roots.

Understanding Roots and Coefficients

Quadratic Equations

A quadratic equation takes the standard form:

$$\ ax^2 + bx + c = 0 $$

where $a$, $b$, and $c$ are coefficients. According to Vieta's formulas, the sum and product of the roots ($r_1$ and $r_2$) can be expressed as:

Sum of roots: $r_1 + r_2 = -\frac{b}{a}$
Product of roots: $r_1 \cdot r_2 = \frac{c}{a}$

Example: Let's consider the equation $2x^2 - 4x + 2 = 0$. Here, $a = 2$, $b = -4$, and $c = 2$.

The sum of the roots $r_1 + r_2 = -\frac{-4}{2} = 2$.
The product of the roots $r_1 \cdot r_2 = \frac{2}{2} = 1$.

Cubic Equations

Cubic equations are generally expressed as:

$$\ ax^3 + bx^2 + cx + d = 0 $$

In this case, Vieta's formulas extend to three roots ($r_1$, $r_2$, $r_3$):

Sum of roots: $r_1 + r_2 + r_3 = -\frac{b}{a}$
Sum of products of roots taken two at a time: $r_1 r_2 + r_1 r_3 + r_2 r_3 = \frac{c}{a}$
Product of roots: $r_1 r_2 r_3 = -\frac{d}{a}$

Example: For the equation $x^3 - 6x^2 + 11x - 6 = 0$:

The sum of the roots: $r_1 + r_2 + r_3 = -\frac{-6}{1} = 6$.
The sum of products of roots taken two at a time: $r_1 r_2 + r_1 r_3 + r_2 r_3 = \frac{11}{1} = 11$.
The product of the roots: $r_1 r_2 r_3 = -\frac{-6}{1} = 6$.

Quartic Equations

Quartic equations follow the form:

$$\ ax^4 + bx^3 + cx^2 + dx + e = 0 $$

For four roots ($r_1$, $r_2$, $r_3$, $r_4$), the relationships are:

Sum of roots: $r_1 + r_2 + r_3 + r_4 = -\frac{b}{a}$
Sum of products of roots taken two at a time: $r_1 r_2 + r_1 r_3 + r_1 r_4 + r_2 r_3 + r_2 r_4 + r_3 r_4 = \frac{c}{a}$
Sum of products taken three at a time: $r_1 r_2 r_3 + r_1 r_2 r_4 + r_1 r_3 r_4 + r_2 r_3 r_4 = -\frac{d}{a}$
Product of roots: $r_1 r_2 r_3 r_4 = \frac{e}{a}$

Example: For the quartic equation $x^4 - 10x^3 + 35x^2 - 50x + 24 = 0$:

Sum of roots: $r_1 + r_2 + r_3 + r_4 = -\frac{-10}{1} = 10$.
Sum of products of roots taken two at a time: $= \frac{35}{1} = 35$.
Sum of products taken three at a time: $= -\frac{-50}{1} = 50$.
Product of roots: $= \frac{24}{1} = 24$.

Forming New Equations

Sometimes, we might need to form a new equation where the roots are related to the original roots. If the roots of a polynomial equation are transformed, the new polynomial can be determined using the relationships we discussed.

For instance, if the roots of a quadratic $ax^2 + bx + c = 0$ are shifted by a constant $k$, the new roots become $r_1 + k$ and $r_2 + k$.

To find the new polynomial equation, you can express the new roots back in the standard polynomial form or use the transformations in conjunction with Vieta's formulas.

Example: If the original roots of $x^2 - 5x + 6 = 0$ are $3$ and $2$, and we want to form a new equation where the roots are each decreased by $1$, the new roots will be $2$ and $1$. Thus, the new polynomial will be:

$$ (x - 2)(x - 1) = x^2 - 3x + 2 $$

Symmetric Functions of Roots

Symmetric functions are expressions that remain unchanged when the roots are permuted. Some key functions are:

$\Sigma \alpha = r_1 + r_2 + r_3 + ...$
$\Sigma \alpha \beta = r_1 r_2 + r_1 r_3 + ... + r_n r_n$
$\Sigma \alpha^2 = r_1^2 + r_2^2 + ... + r_n^2$

These can be expressed using the root-coefficient relationships!

Example: For the quadratic equation $x^2 - 4x + 4 = 0$ (where roots are both $2$), we have:

$r_1 + r_2 = 4$
$r_1 r_2 = 4$
Thus, $\Sigma \alpha^2 = r_1^2 + r_2^2 = (r_1 + r_2)^2 - 2r_1 r_2 = 4^2 - 2 \cdot 4 = 16 - 8 = 8$.

Conclusion

In this lesson, we explored the foundational concepts of roots in polynomial equations, focusing on the relationships between roots and coefficients. We also learned how to form new equations based on transformations of the roots and evaluated symmetric functions without directly finding the roots. Mastery of these concepts is crucial for a solid understanding of advanced mathematics and will prove advantageous in future coursework.

Study Notes

The sum of the roots for quadratics, cubics, and quartics is linked directly to coefficients.
Vieta’s formulas simplify calculating relationships between roots.
Transforming roots can yield new polynomial equations.
Symmetric functions capture key information about the roots without needing to solve for them.
Practice these concepts with various polynomial equations to gain fluency.