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SSC MTS Polynomials

Study Material — 14 PYQs (2018–2018) · Concept Notes · Shortcuts

SSC MTS Polynomials is a frequently tested subtopic — 14 previous year questions from 2018–2018 papers are included below with concept notes, key rules and shortcut tricks.

14 PYQs
2018–2018
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Previous Year Questions

SSC MTS Polynomials — Past Exam Questions

14 questions from actual SSC MTS papers · all shown free · click option to reveal solution

Exam Q 12018Previous Year Pattern

If (x + 3) is a factor of the polynomial p(x) = x² + 5x + 6, then p(-3) should equal:

Exam Q 22018Previous Year Pattern

Simplify: (x + 4)(x - 4) using the difference of squares formula.

Exam Q 32018Previous Year Pattern

Find the remainder when p(x) = x³ + 2x² - 5x + 3 is divided by (x - 1).

Exam Q 42018Previous Year Pattern

What is the degree of the polynomial 4x⁵ + 3x³ - 2x + 7?

Exam Q 52018Previous Year Pattern

If (x - 3) is a factor of the polynomial q(x) = x³ - 6x² + 11x - 6, what is the remainder when q(x) is divided by (x - 3)?

Exam Q 62018Previous Year Pattern

The polynomial p(x) = x² - 7x + k has roots α and β. If α + β = 7 and αβ = 10, find the value of k.

Exam Q 72018Previous Year Pattern

If the polynomial r(x) = x³ + ax² + bx + 6 has (x - 1) and (x + 2) as factors, find the value of a + b.

Exam Q 82018Previous Year Pattern

The polynomial s(x) = 2x³ - 3x² - 8x + 12 can be factored as (x - 2)(2x² + px + q). Find the value of p + q.

Exam Q 92018Previous Year Pattern

If the polynomial t(x) = x⁴ - 5x³ + 5x² + 5x - 6 has (x - 1) as a factor, and when divided by (x - 1), the quotient is a cubic polynomial u(x), what is the sum of coefficients of u(x)?

Exam Q 102018Previous 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 112018Previous Year Pattern

If α, β, γ are the roots of x³ - 6x² + 11x - 6 = 0, find the value of (1/α) + (1/β) + (1/γ).

Exam Q 122018Previous Year Pattern

If the polynomial x⁴ + px³ + qx² + rx + s has roots 1, 2, 3, and 4, find the value of p + q + r + s.

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 find the value of (α + γ)(β + δ) + (α + δ)(β + γ).

Exam Q 142018Previous Year Pattern

If p(x) = x³ - 6x² + 11x - 6 and q(x) = x² - 3x + 2, find the remainder when p(x) is divided by q(x).

Concept Notes

Polynomials— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

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.

2.Exam Patterns

What 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

Example

P(x) = x² + 3x + 2 divided by (x - 1). Remainder = P(1) = 1 + 3 + 2 = 6.

3.Worked Example

Solve 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

4.Exam Traps

Common 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.
←Algebraic Identities
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