Study Material — 16 PYQs (2018–2018) · Concept Notes · Shortcuts
SSC CPO Polynomials is a frequently tested subtopic — 16 previous year questions from 2018–2018 papers are included below with concept notes, key rules and shortcut tricks.
16 questions from actual SSC CPO papers · all shown free · click option to reveal solution
Exam Q 12018Previous Year Pattern
What is the degree of the polynomial 4x⁵ + 3x³ - 2x + 7?
Exam Q 22018Previous Year Pattern
If (x + 3) is a factor of the polynomial p(x) = x² + 5x + 6, what is the value of p(-3)?
Exam Q 32018Previous Year Pattern
If p(x) = x³ - 6x² + 11x - 6 and (x - 1) is a factor, find the other two factors.
Exam Q 42018Previous Year Pattern
What is the remainder when p(x) = 2x³ + 3x² - 5x + 4 is divided by (x - 1)?
Exam Q 52018Previous Year Pattern
Simplify: (x + 2)(x - 2) - (x + 1)²
Exam Q 62018Previous Year Pattern
Divide the polynomial s(x) = x³ + 2x² - 5x - 6 by (x + 1) and find the remainder.
Exam Q 72018Previous Year Pattern
The polynomial t(x) = x³ - 4x² + ax + b is divisible by both (x - 1) and (x - 2). Find the value of a + b.
Exam Q 82018Previous Year Pattern
If (x - 3) is a factor of the polynomial q(x) = x³ - 6x² + 11x - 6, what is the value of q(3)?
Exam Q 92018Previous Year Pattern
If the polynomial r(x) = x² + bx + 12 has roots that are 3 and 4, find the value of b.
Exam Q 102018Previous Year Pattern
If p(x) = x² - 7x + 10 and q(x) = x - 2, find the quotient when p(x) is divided by q(x).
Exam Q 112018Previous Year Pattern
A cubic polynomial p(x) = x³ + ax² + bx + c has roots α, β, γ. If α + β + γ = 6, αβ + βγ + γα = 11, and αβγ = 6, then the value of p(2) is:
Exam Q 122018Previous Year Pattern
If p(x) = x³ - 6x² + 11x - 6 and q(x) = x² - 3x + 2, then the remainder when p(x) is divided by q(x) is:
Exam Q 132018Previous Year Pattern
If α and β are roots of x² - 5x + 6 = 0, and γ and δ are roots of x² - 7x + 12 = 0, then the value of (α + β + γ + δ)² - 4(αβ + γδ) is:
Exam Q 142018Previous Year Pattern
A polynomial p(x) when divided by (x - 2) leaves remainder 5, and when divided by (x - 3) leaves remainder 7. If p(x) is divided by (x - 2)(x - 3), the remainder is ax + b. Find a + b:
Exam Q 152018Previous Year Pattern
If the polynomial x⁴ + px³ + qx² + rx + s has roots in arithmetic progression with common difference 2, and the sum of roots is 8, then the product of the first and last roots is:
Exam Q 162018Previous Year Pattern
If p(x) = x³ - 3x² + kx - 2 has (x - 1) as a factor, and q(x) = x² - 2x + m has (x - 1) as a factor, then the value of k + m is:
Concept Notes
Polynomials— Rules & Concept
Core ConceptRead this first — the foundation of the topic
CORE CONCEPT
A polynomial in one variable x looks like: ax^n + bx^(n-1) + ... + cx + d, where a, b, c, d are constants (called coefficients) and n is a whole number (called the degree). For example, 3x² + 5x + 2 is a polynomial of degree 2
KEY RULES
The DEGREE is the highest power of the variable. In 4x³ + 2x + 1, degree = 3.
2. The LEADING COEFFICIENT is the coefficient of the highest degree term. In 4x³ + 2x + 1, leading coefficient = 4.
3.
The CONSTANT TERM is the term without any variable. In 4x³ + 2x + 1, constant = 1.
4. A polynomial can have multiple variables: 3x²y + 2xy + 5 is valid.
5
Remainder Theorem
If polynomial P(x) is divided by (x - a), the remainder equals P(a).
6
Factor Theorem
(x - a) is a factor of P(x) if and only if P(a) = 0.
Exam PatternsWhat examiners ask — read before attempting PYQs
SSC CGL typically asks
- Finding remainders using Remainder Theorem
- Identifying if an expression is a polynomial
- Finding the degree and coefficients
- Factorizing polynomials
- Finding roots/zeros of polynomials
SHORTCUT:
To find remainder when P(x) is divided by (x - a): Simply substitute x = a in P(x). Don't do actual division
Worked ExampleSolve this step-by-step before moving on
Question: Find the remainder when 2x³ - 5x² + 4x - 3 is divided by (x - 2).
Solution:
Using Remainder Theorem, substitute x = 2:
P(2) = 2(2)³ - 5(2)² + 4(2) - 3
= 2(8) - 5(4) + 8 - 3
= 16 - 20 + 8 - 3
= 1
Remainder = 1
Exam TrapsCommon mistakes students make — avoid these
Students confuse "polynomial" with any algebraic expression. Remember: 1/x + 2, √x + 3, or x^(-2) are NOT polynomials because they have negative or fractional powers, or division by variables.
Key Points to Remember
Polynomial = expression with variables and constants using only addition, subtraction, and multiplication (no division by variables).
Degree = the highest power of the variable in the polynomial.
Remainder Theorem: Remainder when P(x) is divided by (x-a) equals P(a).
Factor Theorem: (x-a) is a factor of P(x) if P(a) = 0.
Leading coefficient = coefficient of the term with highest degree.
To check if expression is a polynomial: all powers must be non-negative whole numbers.
Exam-Specific Tips
Remainder Theorem states: If P(x) is divided by (x - a), remainder = P(a).
Factor Theorem states: (x - a) is a factor of P(x) ⟺ P(a) = 0.
The degree of a polynomial is the highest power of the variable present.
The constant term of a polynomial P(x) equals P(0).
If P(x) has degree n, then P(x) ÷ (x - a) gives quotient of degree (n-1) and remainder of degree 0.
A polynomial cannot have variables in the denominator or have negative/fractional exponents.
The sum or product of two polynomials is always a polynomial.
A polynomial of degree n has at most n real roots/zeros.
60-Second Revision — Polynomials
Remember: Remainder Theorem saves time—just substitute x = a in P(x) instead of doing long division.
Formula: P(x) ÷ (x - a) gives remainder P(a); use this for all remainder questions.
Trap: Not all algebraic expressions are polynomials—check that all powers are non-negative whole numbers.
Factor Theorem: If P(a) = 0, then (x - a) is a factor; use this to find factors quickly.
Degree of P(x) = highest power; leading coefficient = coefficient of that term.
Quick check: 1/x, √x, x^(-1) are NOT polynomials; 3x² + 5x + 2 IS a polynomial.
For factorization: try simple values like 1, -1, 2, -2 first using Factor Theorem to find one factor, then divide.