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2.1 Algebraic Equations, Polynomials & Complex Numbers
PRC Registered Electrical Engineer REE
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1 Chapter 1: Overview, RA 7920 & Electrical Engineering Ethics
1.1 PRC Board of Electrical Engineering & REE Examination Overview 1.2 Republic Act No. 7920 (New Electrical Engineering Law) & License Classes 1.3 Code of Ethics & Professional Practice for Electrical Engineers
2 Chapter 2: Math — Algebra, Trig & Geometry
2.1 Algebraic Equations, Polynomials & Complex Numbers 2.2 Trigonometry — Identities, Triangles & Spherical Trig 2.3 Analytic & Solid Geometry
3 Chapter 3: Engineering Mathematics — Calculus & Differential Equations
3.1 Differential Calculus, Limits, Derivatives & Applications 3.2 Integral Calculus, Integration Techniques & Area/Volume Applications 3.3 Ordinary Differential Equations & Initial-Value Problems
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6.1 Numerical Root-Finding & Solution of Linear Systems 6.2 Interpolation, Numerical Integration & Numerical Solution of ODEs
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8.1 Engineering Mechanics — Statics & Dynamics 8.2 Strength of Materials & Mechanics of Deformable Bodies 8.3 Thermal-Fluid Engineering — Thermodynamics & Fluid Mechanics
9 Chapter 9: ESAS — Engineering Economics, Management & Contracts
9.1 Engineering Economics & Depreciation 9.2 Engineering Management & Ethics 9.3 Philippine Electrical Engineering Law, Code & Contracts 9.4 Technopreneurship 101 & Management of Engineering Projects
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10.1 PEC Fundamentals, Safety Standards & Grounding/Bonding 10.2 Conductor Sizing, Ampacity Adjustment & Voltage Drop 10.3 Overcurrent Protection, Branch Circuit & Feeder Sizing 10.4 Basic Occupational Safety & Health (BOSH) and Electrical Safety Standards
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11.1 PEC Equipment Installation & Wiring Methods 11.2 Utility Systems, Transformers & Switchgear 11.3 Special Occupancies & Hazardous Locations
12 Chapter 12: Electrical Professional — DC & AC Circuit Fundamentals
12.1 DC Circuit Analysis & Network Theorems 12.2 Single-Phase AC Circuits & Phasor Analysis 12.3 Balanced & Unbalanced Three-Phase AC Circuits
13 Chapter 13: Electrical Professional — EM Fields & Transients
13.1 Electrostatics, Magnetostatics & Maxwell's Equations 13.2 Electromagnetic Waves & Transmission Line Fields 13.3 First-Order & Second-Order Circuit Transients
14 Chapter 14: Electrical Professional — Electronics, Industrial Electronics & Communications
14.1 Electronics 1 & 2 — Semiconductor Devices & Analog Circuits 14.2 Electrical Apparatus & Devices — Industrial Electronics & Power Converters 14.3 Fundamentals of Electronic Communications
15 Chapter 15: Electrical Professional — Transformers & DC Machines
15.1 Single-Phase & Three-Phase Transformers 15.2 Transformer Connections & Special Transformers 15.3 DC Generators & DC Motors
16 Chapter 16: Electrical Professional — AC Synchronous & Induction Machinery
16.1 Synchronous Generators (Alternators) 16.2 Synchronous Motors & Special AC Machines 16.3 Three-Phase & Single-Phase Induction Motors
17 Chapter 17: Electrical Professional — Power Plants & Energy Conversion
17.1 Hydroelectric, Thermal & Combined Cycle Power Plants 17.2 Renewable Energy Systems & Geothermal Power Plants 17.3 Power Plant Economics, Load Curves & Energy Conversion
18 Chapter 18: Electrical Professional — Transmission & Distribution
18.1 Transmission Line Parameters & Performance Models 18.2 Mechanical Overhead Line Design, Underground Cables & Insulators 18.3 Distribution Systems, Substation Layouts & Power Flow Control
19 Chapter 19: Electrical Professional — Power System Analysis
19.1 The Per-Unit System, Base Conversion & Impedance Diagrams 19.2 Symmetrical Components & Unbalanced Fault Analysis 19.3 Load Flow, Economic Dispatch & Power System Stability
20 Chapter 20: Electrical Professional — Protection, Control, Instrumentation & Illumination
20.1 Power System Protection & Protective Relaying 20.2 Industrial Motor Control, Logic Circuits & PLC Automation 20.3 Electrical Instrumentation, Transducers & Measurement Systems 20.4 Illumination Engineering, Lighting Design & Photometrics 20.5 Feedback Control Systems & Research Methods
2.1 Algebraic Equations, Polynomials & Complex Numbers
Key Takeaways Explain This Topic AI
The quadratic discriminant D = b^2 - 4ac dictates root characteristics: real distinct (D > 0), real repeated (D = 0), or complex conjugate pairs (D < 0), with Vieta's relations x_1 + x_2 = -b/a and x_1 * x_2 = c/a.
The Remainder Theorem states P(r) is the remainder when P(x) is divided by (x - r), while Vieta's formulas extend root-coefficient relationships to n-th degree polynomials.
Arithmetic (AP), Geometric (GP), and Harmonic (HP) progressions are connected by the fundamental mean inequality and identity GM^2 = AM * HM for positive terms.
The Binomial Theorem enables rapid evaluation of ( a + b ) n (a + b)^n ( a + b ) n expansions and specific term extraction via T r = ( n r − 1 ) a n − r + 1 b r − 1 T_r = \binom{n}{r-1} a^{n-r+1} b^{r-1} T r = ( r − 1 n ) a n − r + 1 b r − 1 .
In electrical engineering, complex numbers represent AC phasors and impedances, utilizing Euler's identity e j θ = cos θ + j sin θ e^{j\theta} = \cos\theta + j\sin\theta e j θ = cos θ + j sin θ and De Moivre's Theorem z n = r n ∠ ( n θ ) z^n = r^n \angle (n\theta) z n = r n ∠ ( n θ ) for calculations.
Last updated: August 2026
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.
1. Quadratic Equations & Roots
A quadratic equation is a second-degree polynomial equation in standard form: a x 2 + b x + c = 0 ( a ≠ 0 ) ax^2 + bx + c = 0 \quad (a \neq 0) a x 2 + b x + c = 0 ( a = 0 )
The Quadratic Formula & Discriminant
The roots x 1 , x 2 x_1, x_2 x 1 , x 2 are given by: x = − b ± b 2 − 4 a c 2 a x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} x = 2 a − b ± b 2 − 4 a c
The term under the radical, D = b 2 − 4 a c D = b^2 - 4ac D = b 2 − 4 a c , is the discriminant . It dictates the nature of the roots:
- Discriminant Value Nature of Roots Graphical Interpretation
- D > 0 D > 0 D > 0 Two real, distinct roots Curve intersects the x-axis at two distinct points
- D = 0 D = 0 D = 0 One real repeated root (double root) Curve is tangent to the x-axis at one point
- D < 0 D < 0 D < 0 Two complex conjugate roots ( a ± j b a \pm j b a ± jb ) Curve does not intersect the x-axis
- Vieta's Formulas for Quadratic Equations
For any quadratic equation a x 2 + b x + c = 0 ax^2 + bx + c = 0 a x 2 + b x + c = 0 with roots x 1 x_1 x 1 and x 2 x_2 x 2 : Sum of roots: x 1 + x 2 = − b a \text{Sum of roots: } x_1 + x_2 = -\frac{b}{a} Sum of roots: x 1 + x 2 = − a b Product of roots: x 1 ⋅ x 2 = c a \text{Product of roots: } x_1 \cdot x_2 = \frac{c}{a} Product of roots: x 1 ⋅ x 2 = a c Sum of squares of roots: x 1 2 + x 2 2 = ( x 1 + x 2 ) 2 − 2 x 1 x 2 = ( − b a ) 2 − 2 ( c a ) = b 2 − 2 a c a 2 \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} Sum of squares of roots: x 1 2 + x 2 2 = ( x 1 + x 2 ) 2 − 2 x 1 x 2 = ( − a b ) 2 − 2 ( a c ) = a 2 b 2 − 2 a c
Key Board Exam Application: If a quadratic equation has roots x 1 x_1 x 1 and x 2 x_2 x 2 , the equation can be reconstructed as: x 2 − ( x 1 + x 2 ) x + ( x 1 x 2 ) = 0 x^2 - (x_1 + x_2)x + (x_1 x_2) = 0 x 2 − ( x 1 + x 2 ) x + ( x 1 x 2 ) = 0
- 2. Polynomial Theorems & Higher-Degree Equations
- Remainder Theorem & Factor Theorem
Remainder Theorem: If a polynomial P ( x ) P(x) P ( x ) is divided by ( x − r ) (x - r) ( x − r ) , the remainder is R = P ( r ) R = P(r) R = P ( r ) .
Factor Theorem: A linear expression ( x − r ) (x - r) ( x − r ) is a factor of P ( x ) P(x) P ( x ) if and only if P ( r ) = 0 P(r) = 0 P ( r ) = 0 .
Rational Root Theorem & Synthetic Division
For a polynomial with integer coefficients a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 = 0 a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0 a n x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 = 0 , any rational root x = p / q x = p/q x = p / q must satisfy:
p p p is an integer factor of the constant term a 0 a_0 a 0 .
q q q is an integer factor of the leading coefficient a n a_n a n .
Vieta's Formulas for Cubic Equations
For a cubic equation a x 3 + b x 2 + c x + d = 0 a x^3 + b x^2 + c x + d = 0 a x 3 + b x 2 + c x + d = 0 with roots x 1 , x 2 , x 3 x_1, x_2, x_3 x 1 , x 2 , x 3 : x 1 + x 2 + x 3 = − b a x_1 + x_2 + x_3 = -\frac{b}{a} x 1 + x 2 + x 3 = − a b x 1 x 2 + x 2 x 3 + x 3 x 1 = c a x_1 x_2 + x_2 x_3 + x_3 x_1 = \frac{c}{a} x 1 x 2 + x 2 x 3 + x 3 x 1 = a c x 1 x 2 x 3 = − d a x_1 x_2 x_3 = -\frac{d}{a} x 1 x 2 x 3 = − a d
- 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 d d to the preceding term.
n n n -th term: a n = a 1 + ( n − 1 ) d a_n = a_1 + (n - 1)d a n = a 1 + ( n − 1 ) d
Sum of n n n terms: S n = n 2 [ 2 a 1 + ( n − 1 ) d ] = n 2 ( a 1 + a n ) S_n = \frac{n}{2} [2a_1 + (n - 1)d] = \frac{n}{2} (a_1 + a_n) S n = 2 n [ 2 a 1 + ( n − 1 ) d ] = 2 n ( a 1 + a n )
Arithmetic Mean (AM): The arithmetic mean between two numbers a a a and b b b is AM = a + b 2 \text{AM} = \frac{a + b}{2} AM = 2 a + b .
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 r r .
n n n -th term: a n = a 1 r n − 1 a_n = a_1 r^{n-1} a n = a 1 r n − 1
Sum of finite n n n terms ( r ≠ 1 r \neq 1 r = 1 ): S n = a 1 ( 1 − r n ) 1 − r = a 1 ( r n − 1 ) r − 1 S_n = \frac{a_1 (1 - r^n)}{1 - r} = \frac{a_1 (r^n - 1)}{r - 1} S n = 1 − r a 1 ( 1 − r n ) = r − 1 a 1 ( r n − 1 )
Sum of an Infinite GP ( ∣ r ∣ < 1 |r| < 1 ∣ r ∣ < 1 ): S ∞ = a 1 1 − r S_\infty = \frac{a_1}{1 - r} S ∞ = 1 − r a 1
Geometric Mean (GM): The geometric mean between two positive numbers a a a and b b b is GM = a b \text{GM} = \sqrt{ab} GM = 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 a a and b b b is: HM = 2 a b a + b \text{HM} = \frac{2ab}{a + b} HM = a + b 2 ab
Fundamental Relationship Between Means
For any two positive unequal numbers a a a and b b b : AM > GM > HM \text{AM} > \text{GM} > \text{HM} AM > GM > HM GM 2 = AM ⋅ HM \text{GM}^2 = \text{AM} \cdot \text{HM} GM 2 = AM ⋅ HM
4. Binomial Theorem
The Binomial Theorem provides the algebraic expansion of powers of a binomial ( a + b ) n (a + b)^n ( a + b ) n : ( a + b ) n = ∑ k = 0 n ( n k ) a n − k b k = ( n 0 ) a n + ( n 1 ) a n − 1 b + ( n 2 ) a n − 2 b 2 + ⋯ + ( n n ) 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 ( a + b ) n = k = 0 ∑ n ( k n ) a n − k b k = ( 0 n ) a n + ( 1 n ) a n − 1 b + ( 2 n ) a n − 2 b 2 + ⋯ + ( n n ) b n
where the binomial coefficient is: ( n k ) = C ( n , k ) = n ! k ! ( n − k ) ! \binom{n}{k} = C(n,k) = \frac{n!}{k!(n - k)!} ( k n ) = C ( n , k ) = k ! ( n − k )! n !
Properties of Binomial Expansion
Total Number of Terms: The expansion of ( a + b ) n (a + b)^n ( a + b ) n contains n + 1 n + 1 n + 1 terms.
The r r r -th Term Formula: The r r r -th term T r T_r T r in the expansion of ( a + b ) n (a + b)^n ( a + b ) n is: T r = ( n r − 1 ) a n − r + 1 b r − 1 T_r = \binom{n}{r - 1} a^{n - r + 1} b^{r - 1} T r = ( r − 1 n ) a n − r + 1 b r − 1
Sum of Coefficients: Setting a = 1 a = 1 a = 1 and b = 1 b = 1 b = 1 , the sum of all binomial coefficients is: ∑ k = 0 n ( n k ) = 2 n \sum_{k=0}^{n} \binom{n}{k} = 2^n k = 0 ∑ n ( k n ) = 2 n
5. Complex Numbers & Phasor Algebra in Electrical Engineering
In electrical engineering, the imaginary unit is denoted by j = − 1 j = \sqrt{-1} j = − 1 to avoid confusion with electric current i i i .
Three Equivalent Representations
Form Expression Conversion Formulas
Rectangular Form z = a + j b z = a + j b z = a + jb a = r cos θ a = r \cos\theta a = r cos θ , b = r sin θ b = r \sin\theta b = r sin θ
Polar Form z = r ∠ θ z = r \angle \theta z = r ∠ θ r = a 2 + b 2 r = \sqrt{a^2 + b^2} r = a 2 + b 2 , θ = arctan ( b a ) \theta = \arctan\left(\frac{b}{a}\right) θ = arctan ( a b )
Exponential Form z = r e j θ z = r e^{j\theta} z = r e j θ Euler's Identity: e j θ = cos θ + j sin θ e^{j\theta} = \cos\theta + j \sin\theta e j θ = cos θ + j sin θ
Mathematical Operations on Complex Numbers
Addition / Subtraction (Rectangular Only): ( a 1 + j b 1 ) ± ( a 2 + j b 2 ) = ( a 1 ± a 2 ) + j ( b 1 ± b 2 ) (a_1 + j b_1) \pm (a_2 + j b_2) = (a_1 \pm a_2) + j (b_1 \pm b_2) ( a 1 + j b 1 ) ± ( a 2 + j b 2 ) = ( a 1 ± a 2 ) + j ( b 1 ± b 2 )
Multiplication (Polar Preferred): ( r 1 ∠ θ 1 ) ⋅ ( r 2 ∠ θ 2 ) = ( r 1 r 2 ) ∠ ( θ 1 + θ 2 ) (r_1 \angle \theta_1) \cdot (r_2 \angle \theta_2) = (r_1 r_2) \angle (\theta_1 + \theta_2) ( r 1 ∠ θ 1 ) ⋅ ( r 2 ∠ θ 2 ) = ( r 1 r 2 ) ∠ ( θ 1 + θ 2 )
Division (Polar Preferred): r 1 ∠ θ 1 r 2 ∠ θ 2 = ( r 1 r 2 ) ∠ ( θ 1 − θ 2 ) \frac{r_1 \angle \theta_1}{r_2 \angle \theta_2} = \left(\frac{r_1}{r_2}\right) \angle (\theta_1 - \theta_2) r 2 ∠ θ 2 r 1 ∠ θ 1 = ( r 2 r 1 ) ∠ ( θ 1 − θ 2 )
De Moivre's Theorem (Powers & Roots): z n = ( r ∠ θ ) n = r n ∠ ( n θ ) = r n [ cos ( n θ ) + j sin ( n θ ) ] z^n = (r \angle \theta)^n = r^n \angle (n\theta) = r^n [\cos(n\theta) + j \sin(n\theta)] z n = ( r ∠ θ ) n = r n ∠ ( n θ ) = r n [ cos ( n θ ) + j sin ( n θ )] Roots: z 1 / n = r 1 / n ∠ ( θ + 2 k π n ) for k = 0 , 1 , 2 , … , n − 1 \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 Roots: z 1/ n = r 1/ n ∠ ( n θ + 2 kπ ) for k = 0 , 1 , 2 , … , n − 1
- Solved Practice Examples
- Example 1: Quadratic Equations & Vieta's Application
Problem: Find the value of k k k such that the roots of 2 x 2 − k x + 8 = 0 2x^2 - kx + 8 = 0 2 x 2 − k x + 8 = 0 satisfy x 1 = 2 x 2 x_1 = 2 x_2 x 1 = 2 x 2 .
Solution:
By Vieta's formulas: x 1 + x 2 = k 2 , x 1 ⋅ x 2 = 8 2 = 4 x_1 + x_2 = \frac{k}{2}, \quad x_1 \cdot x_2 = \frac{8}{2} = 4 x 1 + x 2 = 2 k , x 1 ⋅ x 2 = 2 8 = 4
Substitute x 1 = 2 x 2 x_1 = 2 x_2 x 1 = 2 x 2 into the product equation: ( 2 x 2 ) ⋅ x 2 = 4 ⟹ 2 x 2 2 = 4 ⟹ x 2 2 = 2 ⟹ x 2 = ± 2 (2 x_2) \cdot x_2 = 4 \implies 2 x_2^2 = 4 \implies x_2^2 = 2 \implies x_2 = \pm \sqrt{2} ( 2 x 2 ) ⋅ x 2 = 4 ⟹ 2 x 2 2 = 4 ⟹ x 2 2 = 2 ⟹ x 2 = ± 2
Find x 1 x_1 x 1 : x 1 = 2 ( ± 2 ) = ± 2 2 x_1 = 2 (\pm \sqrt{2}) = \pm 2\sqrt{2} x 1 = 2 ( ± 2 ) = ± 2 2
Calculate k k k : k 2 = x 1 + x 2 = 2 2 + 2 = 3 2 ⟹ k = 6 2 ≈ 8.485 \frac{k}{2} = x_1 + x_2 = 2\sqrt{2} + \sqrt{2} = 3\sqrt{2} \implies k = 6\sqrt{2} \approx 8.485 2 k = x 1 + x 2 = 2 2 + 2 = 3 2 ⟹ k = 6 2 ≈ 8.485
Example 2: Complex Impedance & De Moivre's Powers
Problem: An electrical impedance is given by Z = 3 + j 4 Ω Z = 3 + j4\ \Omega Z = 3 + j 4 Ω . Compute Z 4 Z^4 Z 4 in polar and rectangular form.
Solution:
Convert Z Z Z to polar form: r = 3 2 + 4 2 = 5 r = \sqrt{3^2 + 4^2} = 5 r = 3 2 + 4 2 = 5 θ = arctan ( 4 3 ) ≈ 53.1301 ∘ \theta = \arctan\left(\frac{4}{3}\right) \approx 53.1301^\circ θ = arctan ( 3 4 ) ≈ 53.130 1 ∘ Z = 5 ∠ 53.1301 ∘ Ω Z = 5 \angle 53.1301^\circ\ \Omega Z = 5∠53.130 1 ∘ Ω
Apply De Moivre's Theorem for n = 4 n = 4 n = 4 : Z 4 = 5 4 ∠ ( 4 × 53.1301 ∘ ) = 625 ∠ 212.5204 ∘ Z^4 = 5^4 \angle (4 \times 53.1301^\circ) = 625 \angle 212.5204^\circ Z 4 = 5 4 ∠ ( 4 × 53.130 1 ∘ ) = 625∠212.520 4 ∘
Convert back to rectangular form: a = 625 cos ( 212.5204 ∘ ) = 625 ( − 0.8432 ) = − 527 a = 625 \cos(212.5204^\circ) = 625 (-0.8432) = -527 a = 625 cos ( 212.520 4 ∘ ) = 625 ( − 0.8432 ) = − 527 b = 625 sin ( 212.5204 ∘ ) = 625 ( − 0.5376 ) = − 336 b = 625 \sin(212.5204^\circ) = 625 (-0.5376) = -336 b = 625 sin ( 212.520 4 ∘ ) = 625 ( − 0.5376 ) = − 336 Z 4 = − 527 − j 336 Z^4 = -527 - j336 Z 4 = − 527 − j 336