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Agniveer Army CEE Polynomials

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This page covers Agniveer Army CEE Polynomials with complete concept notes, 8 graded practice MCQs, key points and exam-specific tips. Free to study.

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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

Example

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

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.
Practice MCQs

Polynomials — Practice Questions

8graded MCQs · easy to hard · full solution & trap analysis

All MCQs →
Practice 1easy

If p(x) = 3x² + 5x - 2, find the value of p(2).

Practice 2easy

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

Practice 3easy

If (x - 3) is a factor of the polynomial q(x) = x² - 5x + 6, what is the value of q(3)?

Practice 4medium

If p(x) = 2x² + 3x - 5, find the value of p(2).

Practice 5medium

What is the degree of the polynomial 5x³ + 2x² - x + 7?

Practice 6medium

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

Practice 7medium

Find the remainder when p(x) = x² + 4x + 3 is divided by (x + 1).

Practice 8hard

If p(x) = x³ - 6x² + 11x - 6, and it is known that (x - 1) is a factor of p(x), find the other two linear factors.

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.
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