Algebra

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Algebra — Key Facts for Kenyatta University Quadratic equation: ax² + bx + c = 0; roots = [−b ± √(b² − 4ac)]/2a Sum of roots = −b/a; Product of roots = c/a Factorisation: (x − a)(x − b) = 0 expands to x² − (a+b)x + ab = 0 Partial fractions: decompose rational functions for integration or simplification ⚡ Exam tip: Kenyatta algebra problems often combine quadratic equations with word problems — always identify what the variable represents first!

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🟡 Standard — Core Study

Standard content for students with a few days to months.

Algebra — Kenyatta University Study Guide

Quadratic equations:

• Standard form: ax² + bx + c = 0
• Discriminant D = b² − 4ac
• D > 0: two distinct real roots
• D = 0: equal real roots
• D < 0: complex conjugate roots (no real solutions)

Vieta’s formulas:

• Sum of roots α + β = −b/a
• Product of roots αβ = c/a
• Useful for forming equations from given roots

Factor theorem: If f(a) = 0, then (x − a) is a factor of f(x) Remainder when dividing by (x − a) is f(a)

Partial fractions: For proper fraction (numerator degree < denominator degree):

• Distinct linear factors: A/(x−a) + B/(x−b)
• Repeated linear: A/(x−a) + B/(x−a)²
• Quadratic factor: (Ax+B)/(x²+bx+c)

Arithmetic Progression (AP):

• nth term: a_n = a + (n−1)d
• Sum of n terms: S_n = n/2[2a + (n−1)d] = n(a + l)/2

Geometric Progression (GP):

• nth term: a_n = ar^{n−1}
• Sum of n terms: S_n = a(r^n − 1)/(r − 1), r ≠ 1
• Infinite sum (|r| < 1): S_∞ = a/(1 − r)

Harmonic Progression (HP): Terms are reciprocals of AP nth term: a_n = 1/[1/a + (n−1)d] where d comes from AP of reciprocals

Logarithms:

• log_a(xy) = log_a x + log_a y
• log_a(x/y) = log_a x − log_a y
• log_a(x^n) = n log_a x
• Change of base: log_a x = log_b x/log_b a

Exponentials:

• a^x × a^y = a^{x+y}
• (a^x)^y = a^{xy}
• a^0 = 1; a^1 = a

Binomial expansion: (a + b)^n = Σ C(n,r) a^{n−r} b^r General term: T_{r+1} = C(n,r) a^{n−r} b^r

• Key formula: Roots of ax²+bx+c=0: [−b ± √(b²−4ac)]/2a; sum = −b/a, product = c/a
• Common trap: For equation transformation (roots scaled by k), ensure leading coefficient doesn’t become zero
• Exam weight: 2–3 questions per exam; frequently appears

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🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Algebra — Comprehensive Kenyatta Notes

Nature of roots:

• Both positive: D ≥ 0, α + β > 0, αβ > 0
• Both negative: D ≥ 0, α + β < 0, αβ > 0
• Opposite signs: αβ < 0
• Reciprocal: αβ = 1 → c/a = 1 → c = a

Maximum/minimum of quadratic: For f(x) = ax² + bx + c:

• Vertex at x = −b/2a
• Value = −D/4a = −(b²−4ac)/4a

Transformation of equations:

• Roots multiplied by k: substitute x/k for x
• Roots increased by m: substitute x − m for x
• Reciprocal roots: reverse coefficients

Common roots condition: For ax²+bx+c and dx²+ex+f: One common root: (ae−bd)(bf−ce) = (af−cd)(cb−eb) Both common: a/d = b/e = c/f

Method of differences: If T_n = f(n) − f(n+1), then Σ T_n telescopes Example: 1/[n(n+1)] = 1/n − 1/(n+1); sum = 1 − 1/(N+1)

Summation formulas:

• Σ n = n(n+1)/2
• Σ n² = n(n+1)(2n+1)/6
• Σ n³ = [n(n+1)/2]²

Inequalities:

• AM ≥ GM: (a+b)/2 ≥ √(ab)
• For quadratic f(x) > 0 for all x: a > 0 and D < 0
• For quadratic f(x) ≥ 0: a > 0 and D ≤ 0

Partial fraction decomposition:

1. If degree numerator ≥ degree denominator, divide first
2. Factor denominator completely
3. Set up partial fractions with unknown constants
4. Solve by substituting values or equating coefficients

Logarithmic equations:

• a^{f(x)} = a^{g(x)} → f(x) = g(x) if a > 0, a ≠ 1
• Domain: argument > 0, base > 0, base ≠ 1

Exponential equations:

• If a^x = a^y → x = y (for a > 0, a ≠ 1)
• For different bases, take log: x ln a = y ln b

Surds:

• √ab = √a × √b only if a, b ≥ 0
• Rationalising: 1/(√a + √b) = (√a − √b)/(a − b)

Complex numbers basics: i = √−1; i² = −1 For z = a + ib: Re(z) = a, Im(z) = b |z| = √(a² + b²); z̄ = a − ib

• Remember: quadratic formula: [−b ± √(b²−4ac)]/2a; sum of roots = −b/a; product = c/a; partial fractions: identify factor types first; AM ≥ GM: (a+b)/2 ≥ √(ab)
• Previous years: “Solve 2x² − 5x + 2 = 0” [2023 KU]; “Find sum of infinite GP with a=1, r=1/2” [2024 KU]; “Factorise x³ − 27” [2024 KU]

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📊 Kenyatta University Exam Essentials

Detail	Value
Questions	50 (depending on course)
Time	2–3 hours
Marks	100
Format	Mix of short answer and problem solving

🎯 High-Yield Topics for Kenyatta University Mathematics

• Calculus (Differentiation + Integration) — highest weight
• Algebra (Quadratics, AP/GP/HP) — very high weight
• Trigonometry — high weight
• Statistics and Probability — medium-high weight
• Coordinate Geometry — medium weight

💡 Pro Tips

• Always check discriminant first when determining nature of roots
• For AP/GP/HP problems, identify the type first
• Partial fractions are essential for integration
• For logarithmic equations, always check domain (argument > 0, base > 0 and ≠ 1)

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