2.1 Algebraic Equations, Polynomials & Complex Numbers

2.1 Algebraic Equations, Polynomials & Complex Numbers

Algebraic manipulation, polynomial analysis, and complex number theory form the cornerstone of engineering mathematics for the PRC Registered Electrical Engineer (REE) Licensure Examination. Electrical engineering applications rely heavily on these fundamental mathematical concepts—from solving network loop equations to evaluating AC phasor networks.

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1. Quadratic Equations & Roots

A quadratic equation is a second-degree polynomial equation in standard form:

$$ax^2 + bx + c = 0 \quad (a \neq 0)$$

The Quadratic Formula & Discriminant

The roots $x_1, x_2$ are given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The term under the radical, $D = b^2 - 4ac$, is the discriminant. It dictates the nature of the roots:

Vieta's Formulas for Quadratic Equations

For any quadratic equation $ax^2 + bx + c = 0$ with roots $x_1$ and $x_2$:

$$\text{Sum of roots: } x_1 + x_2 = -\frac{b}{a}$$ $$\text{Product of roots: } x_1 \cdot x_2 = \frac{c}{a}$$ $$\text{Sum of squares of roots: } x_1^2 + x_2^2 = (x_1 + x_2)^2 - 2 x_1 x_2 = \left(-\frac{b}{a}\right)^2 - 2\left(\frac{c}{a}\right) = \frac{b^2 - 2ac}{a^2}$$

Key Board Exam Application: If a quadratic equation has roots $x_1$ and $x_2$, the equation can be reconstructed as: $x^2 - (x_1 + x_2)x + (x_1 x_2) = 0$

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2. Polynomial Theorems & Higher-Degree Equations

Remainder Theorem & Factor Theorem

• Remainder Theorem: If a polynomial $P(x)$ is divided by $(x - r)$, the remainder is $R = P(r)$.
• Factor Theorem: A linear expression $(x - r)$ is a factor of $P(x)$ if and only if $P(r) = 0$.

Rational Root Theorem & Synthetic Division

For a polynomial with integer coefficients $a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0$, any rational root $x = p/q$ must satisfy:

• $p$ is an integer factor of the constant term $a_0$.
• $q$ is an integer factor of the leading coefficient $a_n$.

Vieta's Formulas for Cubic Equations

For a cubic equation $a x^3 + b x^2 + c x + d = 0$ with roots $x_1, x_2, x_3$:

$$x_1 + x_2 + x_3 = -\frac{b}{a}$$ $$x_1 x_2 + x_2 x_3 + x_3 x_1 = \frac{c}{a}$$ $$x_1 x_2 x_3 = -\frac{d}{a}$$

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3. Progressions (Arithmetic, Geometric, Harmonic)

Arithmetic Progression (AP)

An Arithmetic Progression is a sequence where each term after the first is obtained by adding a constant difference $d$ to the preceding term.

• $n$-th term: $a_n = a_1 + (n - 1)d$
• Sum of $n$ terms: $S_n = \frac{n}{2} [2a_1 + (n - 1)d] = \frac{n}{2} (a_1 + a_n)$
• Arithmetic Mean (AM): The arithmetic mean between two numbers $a$ and $b$ is $\text{AM} = \frac{a + b}{2}$.

Geometric Progression (GP)

A Geometric Progression is a sequence where each term after the first is obtained by multiplying the preceding term by a constant ratio $r$.

• $n$-th term: $a_n = a_1 r^{n-1}$
• Sum of finite $n$ terms ($r \neq 1$): $S_n = \frac{a_1 (1 - r^n)}{1 - r} = \frac{a_1 (r^n - 1)}{r - 1}$
• Sum of an Infinite GP ($|r| < 1$): $S_\infty = \frac{a_1}{1 - r}$
• Geometric Mean (GM): The geometric mean between two positive numbers $a$ and $b$ is $\text{GM} = \sqrt{ab}$.

Harmonic Progression (HP)

A Harmonic Progression is a sequence of numbers whose reciprocals form an arithmetic progression.

• Harmonic Mean (HM): The harmonic mean between two positive numbers $a$ and $b$ is: $$\text{HM} = \frac{2ab}{a + b}$$

Fundamental Relationship Between Means

For any two positive unequal numbers $a$ and $b$:

$$\text{AM} > \text{GM} > \text{HM}$$ $$\text{GM}^2 = \text{AM} \cdot \text{HM}$$

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4. Binomial Theorem

The Binomial Theorem provides the algebraic expansion of powers of a binomial $(a + b)^n$:

$$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k = \binom{n}{0} a^n + \binom{n}{1} a^{n-1}b + \binom{n}{2} a^{n-2}b^2 + \dots + \binom{n}{n} b^n$$

where the binomial coefficient is:

$$\binom{n}{k} = C(n,k) = \frac{n!}{k!(n - k)!}$$

Properties of Binomial Expansion

1. Total Number of Terms: The expansion of $(a + b)^n$ contains $n + 1$ terms.
2. The $r$-th Term Formula: The $r$-th term $T_r$ in the expansion of $(a + b)^n$ is: $$T_r = \binom{n}{r - 1} a^{n - r + 1} b^{r - 1}$$
3. Sum of Coefficients: Setting $a = 1$ and $b = 1$, the sum of all binomial coefficients is: $$\sum_{k=0}^{n} \binom{n}{k} = 2^n$$

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5. Complex Numbers & Phasor Algebra in Electrical Engineering

In electrical engineering, the imaginary unit is denoted by $j = \sqrt{-1}$ to avoid confusion with electric current $i$.

Three Equivalent Representations

Mathematical Operations on Complex Numbers

• Addition / Subtraction (Rectangular Only): $$(a_1 + j b_1) \pm (a_2 + j b_2) = (a_1 \pm a_2) + j (b_1 \pm b_2)$$
• Multiplication (Polar Preferred): $$(r_1 \angle \theta_1) \cdot (r_2 \angle \theta_2) = (r_1 r_2) \angle (\theta_1 + \theta_2)$$
• Division (Polar Preferred): $$\frac{r_1 \angle \theta_1}{r_2 \angle \theta_2} = \left(\frac{r_1}{r_2}\right) \angle (\theta_1 - \theta_2)$$
• De Moivre's Theorem (Powers & Roots): $$z^n = (r \angle \theta)^n = r^n \angle (n\theta) = r^n [\cos(n\theta) + j \sin(n\theta)]$$ $$\text{Roots: } z^{1/n} = r^{1/n} \angle \left(\frac{\theta + 2k\pi}{n}\right) \quad \text{for } k = 0, 1, 2, \dots, n-1$$

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Solved Practice Examples

Example 1: Quadratic Equations & Vieta's Application

Problem: Find the value of $k$ such that the roots of $2x^2 - kx + 8 = 0$ satisfy $x_1 = 2 x_2$.

Solution:

1. By Vieta's formulas: $$x_1 + x_2 = \frac{k}{2}, \quad x_1 \cdot x_2 = \frac{8}{2} = 4$$
2. Substitute $x_1 = 2 x_2$ into the product equation: $$(2 x_2) \cdot x_2 = 4 \implies 2 x_2^2 = 4 \implies x_2^2 = 2 \implies x_2 = \pm \sqrt{2}$$
3. Find $x_1$: $$x_1 = 2 (\pm \sqrt{2}) = \pm 2\sqrt{2}$$
4. Calculate $k$: $$\frac{k}{2} = x_1 + x_2 = 2\sqrt{2} + \sqrt{2} = 3\sqrt{2} \implies k = 6\sqrt{2} \approx 8.485$$

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Example 2: Complex Impedance & De Moivre's Powers

Problem: An electrical impedance is given by $Z = 3 + j4\ \Omega$. Compute $Z^4$ in polar and rectangular form.

Solution:

1. Convert $Z$ to polar form: $$r = \sqrt{3^2 + 4^2} = 5$$ $$\theta = \arctan\left(\frac{4}{3}\right) \approx 53.1301^\circ$$ $$Z = 5 \angle 53.1301^\circ\ \Omega$$
2. Apply De Moivre's Theorem for $n = 4$: $$Z^4 = 5^4 \angle (4 \times 53.1301^\circ) = 625 \angle 212.5204^\circ$$
3. Convert back to rectangular form: $$a = 625 \cos(212.5204^\circ) = 625 (-0.8432) = -527$$ $$b = 625 \sin(212.5204^\circ) = 625 (-0.5376) = -336$$ $$Z^4 = -527 - j336$$