Polynomials and Remainder Theorem

What the exams ask here

• ax^2+bx+5 leaves remainders r1,r2 find 2b-4a
• remainder when p(x) divided by (x-1)(x-2)
• which is divisible by 3^10+2
• common factor coefficients
• 4x^10 - x^9 + ... + cx^6 + ... leaves remainder 2 on division by x - 1 -> c + 6 (Indore 2023 SA Q7)
• remainders 2 and 1 on division by x - 1 and x - 2 -> remainder by (x - 1)(x - 2) (Indore 2023 Q26)
• ax^2 + bx + 5 with remainders 3 and 2 -> 2b - 4a (Indore 2025 SA Q4)
• 2x^3 + ax^2 + bx + 3 with factor x - 1 and remainder 15 at x = -2 -> (a, b) (JIPMAT 2023 Q22)
• number of zeros from the graph of y = f(x) (JIPMAT 2023 Q27)
• HCF of x^2 - px - q and 5x^2 - 3px - 15q is x - 3 (JIPMAT 2024 Q27)
• factors of x^3 - 6x^2 + 11x - 6 (JIPMAT 2024 Q28)
• x^4 + ax^3 + (3a - 4)x^2 + 2(a - 1)x - 4 = (x^2 + px + 2)(x^2 + qx - 2): number of a (Indore 2026 MCQ Q7, memory-based)
• GCD / LCM of two polynomials
• which number is divisible by 3^10 + 2 (Indore 2025 MCQ Q5, stored in surds-indices)