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Polynomials

CRPF Constable · Quantitative Aptitude

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← Algebraic Identities

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Core Concept

Read 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.

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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.

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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 COMMON MISTAKE: 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

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 Facts

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

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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