Algebra

🟢 Lite — Quick Review

Rapid summary for last-minute revision before your exam.

Algebra — Key Facts for BUET Polynomial degree n has n roots (real or complex); sum of roots = −b/a, product = c/a Quadratic: ax² + bx + c = 0; roots = [−b ± √(b² − 4ac)]/2a For quadratic with real roots: discriminant D ≥ 0; if D = 0, roots are equal; if D < 0, complex conjugate roots Partial fractions: decompose rational functions to integrate or simplify ⚡ Exam tip: BUET algebra problems often combine with complex numbers or matrices — mastering Vieta’s formulas is essential!

---

🟡 Standard — Core Study

Standard content for students with a few days to months.

Algebra — BUET Study Guide

Quadratic equations:

• Standard form: ax² + bx + c = 0
• Sum of roots (α + β) = −b/a
• Product of roots (αβ) = c/a
• For equal roots: b² = 4ac (D = 0)
• For real roots: b² ≥ 4ac (D ≥ 0)

Nature of roots analysis:

• Both positive: D ≥ 0, α + β > 0, αβ > 0, f(0) > 0
• Both negative: D ≥ 0, α + β < 0, αβ > 0, f(0) > 0
• Opposite signs: αβ < 0 (sufficient)
• Reciprocal: if one root is α, other is 1/α → c = a
• Equal magnitude, opposite sign: b = 0

Transformation of equations:

• Roots scaled by k: equation becomes a(x/k)² + b(x/k) + c = 0 → ak²x² + bkx + c = 0
• Roots shifted by m: substitute x − m → a(x−m)² + b(x−m) + c = 0
• Reciprocal roots: reverse coefficients → cx² + bx + a = 0

Partial fractions: For proper fraction P(x)/Q(x) where degree of P < degree of Q:

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

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
• Sum of coefficients: put a = b = 1 → 2^n
• Middle term(s): if n even, one middle term T_{(n/2)+1}; if n odd, two middle terms

Arithmetic Progression:

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

Geometric Progression:

• 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:

• Terms are reciprocals of AP: if a, b, c are in HP, then 1/a, 1/b, 1/c are in AP
• nth term: a_n = 1/[1/a + (n−1)d] where d is from the 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

• Key formula: Sum of roots = −b/a, Product = c/a; (a+b)^n expansion; partial fraction decomposition rules

• Common trap: For equation transformation, ensure leading coefficient doesn’t become zero — if ak² = 0, the transformed equation becomes linear

• Exam weight: 2–3 questions per exam (8–12 marks); very high weight

---

🔴 Extended — Deep Dive

Comprehensive coverage for students on a longer study timeline.

Algebra — Comprehensive BUET Notes

Condition for common roots: Two quadratics a₁x² + b₁x + c₁ = 0 and a₂x² + b₂x + c₂ = 0:

• One common root: (a₁b₂ − a₂b₁)(b₁c₂ − b₂c₁) = (a₁c₂ − a₂c₁)(c₁b₂ − c₂b₁)
• Both common: a₁/a₂ = b₁/b₂ = c₁/c₂

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

• Vertex at x = −b/2a
• If a > 0: minimum value = −D/4a at x = −b/2a
• If a < 0: maximum value = −D/4a at x = −b/2a

Location of roots:

• Both roots in (m, n): D ≥ 0, f(m) > 0, f(n) > 0, −b/2a ∈ (m, n)
• One root in (m, n), one outside: f(m)·f(n) < 0
• Roots on either side of k: f(k) < 0

Summation of series:

• Sum of squares: 1² + 2² + … + n² = n(n+1)(2n+1)/6
• Sum of cubes: 1³ + 2³ + … + n³ = [n(n+1)/2]²
• Arithmetic-Geometric series: multiply by r and subtract
• Telescoping series: express general term as difference of two terms

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

Exponential and logarithmic equations:

• a^{f(x)} = a^{g(x)} → f(x) = g(x) if a > 0 and a ≠ 1
• If base differs, take log: f(x)^{g(x)} = h(x)^{g(x)} → if bases are unequal but exponents same…

Important inequalities:

• AM ≥ GM: (a + b)/2 ≥ √(ab)
• For any real x: x² + 1 ≥ 2|x|
• Cauchy-Schwarz: (Σ a_i²)(Σ b_i²) ≥ (Σ a_i b_i)²

De Moivre’s theorem: (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) This is used for evaluating expressions like (1 + i)^n

nth roots of unity: For z^n = 1, solutions are z = e^{2πik/n} for k = 0, 1, …, n−1 Sum of all roots = 0

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

Symmetric functions of roots: For polynomial with roots α, β, γ:

• Elementary symmetric: s₁ = α+β+γ, s₂ = αβ+βγ+γα, s₃ = αβγ
• Express symmetric sums in terms of coefficients

Partial fraction cases:

1. Distinct linear factors: A/(x−a) + B/(x−b)
2. Repeated linear: A/(x−a) + B/(x−a)² + C/(x−a)³
3. Irreducible quadratic: (Ax+B)/(x²+bx+c)

Logarithmic inequalities:

• If a > 1, log_a x > log_a y ↔ x > y

• If 0 < a < 1, log_a x > log_a y ↔ x < y

• domain of log: argument > 0

• Remember: Quadratic: sum = −b/a, product = c/a; binomial: general term C(n,r)a^{n−r}b^r; sum of cubes = [n(n+1)/2]²; partial fractions: identify factor types first

• Previous years: “Find range of k for which equation x² − kx + 4 = 0 has real roots” [2023 BUET]; “Sum of infinite GP is 3 and first term is 1, find ratio” [2024 BUET]; “Factorise x³ − 6x² + 11x − 6” [2024 BUET]

---

📊 BUET Admission Exam Essentials

Detail	Value
Questions	Varies by year (~40-50 MCQ)
Time	Usually 2–3 hours
Marks	Varies by section
Subjects	Mathematics (highest weight), Physics, Chemistry
Negative	Usually no negative marking in BUET
Mode	Written + MCQ depending on year

🎯 High-Yield Topics for BUET Mathematics

• Calculus (Differentiation + Integration) — highest weight
• Algebra (Quadratics, AP/GP/HP) — very high weight
• Coordinate Geometry (Circle, Conics) — high weight
• Trigonometry — medium-high weight
• Complex Numbers — medium weight

📝 Previous Year Question Patterns

• Algebra: 3–5 questions per exam, 12–20 marks
• Common patterns: quadratic equations, progression problems, binomial expansion, partial fractions
• Weight: very high — prioritise algebra mastery

💡 Pro Tips

• BUET mathematics is heavily calculus-based — ensure algebra foundations are solid
• Partial fractions are essential for integration — practice decomposition
• Vieta’s formulas for quadratics appear in almost every algebra problem
• For progression problems, identify AP/GP/HP first and use appropriate formulas
• Logarithmic equations: always check domain (argument > 0, base > 0, base ≠ 1)

🔗 Official Resources

• BUET Official
• [BUET Admission Portal](https:// admission.buet.ac.bd)

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.