Class 9 Polynomials

Comprehensive Study Material on Polynomials

For CBSE/ICSE Students

PART 1: Detailed Theory and Exercises

A polynomial in one variable is an algebraic expression consisting of terms involving powers of a single variable. A general polynomial in one variable x is written as:

P(x) = a₀ + a₁x + a₂x² + … + aₙxⁿ

where a₀, a₁, a₂, …, aₙ are real numbers (called coefficients), and n is a non-negative integer representing the degree of the polynomial.

Example: 3x² – 4x + 7 is a polynomial of degree 2.

Solved Examples:

  • 2x³ + 5x² – 3x + 8 is a polynomial in x.
  • x² + 4 is a polynomial of degree 2.
  • 7x⁵ is a polynomial with one term and degree 5.
  • 4x⁻² + x + 1 is not a polynomial (negative power).
  • √x + 3 is not a polynomial (fractional exponent).

Exercise:

  • Identify whether the following are polynomials: x³ – 2x + 1, 3x⁻¹ + 2x, √x + 2, 5x⁴ – x + 9
  • Write a polynomial of degree 4.
  • Write any three polynomials in variable x of degree 2.
  • Write a polynomial with 4 terms.
  • Determine the degree of the polynomial: 2x⁴ + 5x² + 7
  • State whether 4x² + 2/x is a polynomial or not.
  • Write a polynomial with constant term 3 and highest degree 2.
  • Which of the following are not polynomials? x² + 1/x, x¹ᐟ², x³ – 5x + 7

Every polynomial is made up of terms. Each term has a coefficient and a variable part raised to a power. The coefficient is the constant multiplier of the term. The degree of the polynomial is the highest power of the variable.

Example: P(x) = 5x³ – 2x² + 3x – 7

  • Terms: 5x³, -2x², 3x, -7
  • Coefficients: 5, -2, 3, -7
  • Degree: 3

Solved Examples:

  • In 4x⁵ + 3x² – x + 6, the degree is 5, leading coefficient is 4.
  • In -x³ + 7x, the coefficient of x³ is -1, degree is 3.
  • In 2x² – x + 9, there are 3 terms.
  • x – 2 has 2 terms, degree is 1.
  • Constant term of 6x² + 4x – 9 is -9.

Exercise:

  • Find the degree of 3x⁵ – x² + 2x + 1
  • Identify all coefficients in 2x³ – 5x² + x – 4
  • Find the constant term in x⁴ – 3x² + 6
  • State the number of terms in 7x³ – x + 2
  • Write a polynomial with leading coefficient 9 and degree 3
  • What is the coefficient of x² in 4x² + 5x – 1?
  • Identify all terms in -3x⁴ + x² – x + 1
  • Write a polynomial with 5 terms and degree 4

Polynomials are classified by their degree:

Type General Form Degree
Zero Polynomial 0 Undefined or 0
Linear ax + b 1
Quadratic ax² + bx + c 2
Cubic ax³ + bx² + cx + d 3

Solved Examples:

  • 0 is a zero polynomial.
  • x + 5 is a linear polynomial.
  • x² + 4x + 4 is a quadratic polynomial.
  • 2x³ + x² + 7x + 3 is a cubic polynomial.
  • 7 is a constant polynomial.

Exercise:

  • Identify the type of polynomial: 3x + 2
  • Classify x² + x + 1
  • Which type of polynomial is x³ + 2x² + 1?
  • Write a quadratic polynomial.
  • Write a linear polynomial with constant term 4.
  • Identify type of polynomial: 0
  • Is 2x linear or constant?
  • Write one example of a cubic polynomial

Polynomials are also classified based on the number of terms:

  • Monomial: A polynomial with one term (e.g., 5x³)
  • Binomial: A polynomial with two terms (e.g., x + 3)
  • Trinomial: A polynomial with three terms (e.g., x² + 4x + 7)

Solved Examples:

  • 3x² is a monomial.
  • x² + x is a binomial.
  • x³ + 2x + 1 is a trinomial.
  • 7x⁴ is a monomial.
  • x + 4 is a binomial.

Exercise:

  • Classify: 2x³, x + 1, x² – x + 3
  • Which of these is a binomial: 4x, x² + 3x, x³ + x² + x + 1?
  • Write one example each: monomial, binomial, trinomial
  • Identify monomials in x⁴ – 2x² + 5x – 3
  • Classify 3x² + 5x
  • Write two binomials.
  • Write a trinomial with degree 2.
  • Classify 7x³ – x + 1

Polynomials are classified by their degree:

Type General Form Degree
Zero Polynomial 0 Undefined or 0
Linear ax + b 1
Quadratic ax² + bx + c 2
Cubic ax³ + bx² + cx + d 3

Solved Examples:

  • 0 is a zero polynomial.
  • x + 5 is a linear polynomial.
  • x² + 4x + 4 is a quadratic polynomial.
  • 2x³ + x² + 7x + 3 is a cubic polynomial.
  • 7 is a constant polynomial.

Exercise:

  1. Identify the type of polynomial: 3x + 2
  2. Classify x² + x + 1
  3. Which type of polynomial is x³ + 2x² + 1?
  4. Write a quadratic polynomial.
  5. Write a linear polynomial with constant term 4.
  6. Identify type of polynomial: 0
  7. Is 2x linear or constant?
  8. Write one example of a cubic polynomial.

Polynomials are also classified based on the number of terms they contain:

  • Monomial: A polynomial with one term (e.g., 5x³)
  • Binomial: A polynomial with two terms (e.g., x + 3)
  • Trinomial: A polynomial with three terms (e.g., x² + 4x + 7)

Solved Examples:

  • 3x² is a monomial.
  • x² + x is a binomial.
  • x³ + 2x + 1 is a trinomial.
  • 7x⁴ is a monomial.
  • x + 4 is a binomial.

Exercise:

  1. Classify: 2x³, x + 1, x² - x + 3
  2. Which of these is a binomial: 4x, x² + 3x, x³ + x² + x + 1?
  3. Write one example each: monomial, binomial, trinomial
  4. Identify monomials in x⁴ - 2x² + 5x - 3
  5. Classify 3x² + 5x
  6. Write two binomials.
  7. Write a trinomial with degree 2.
  8. Classify 7x³ - x + 1

Operations on polynomials involve combining like terms. The operations include:

  • Addition: Combine like terms.
  • Subtraction: Subtract corresponding terms.
  • Multiplication: Use distributive property (FOIL for binomials).

Solved Examples:

  • (x² + 3x + 2) + (2x² + x - 1) = 3x² + 4x + 1
  • (x² - 2x + 1) - (x² + x + 1) = -3x
  • (x + 2)(x + 3) = x² + 5x + 6
  • (2x - 1)(x + 4) = 2x² + 7x - 4
  • (x² + x)(x - 1) = x³ - x
  • (x - 2)² = x² - 4x + 4

Exercise:

  1. Add: (x + 2) + (2x - 3)
  2. Subtract: (3x² + x - 4) - (x² - x + 2)
  3. Multiply: (x + 1)(x - 1)
  4. Multiply: (2x + 5)(x - 3)
  5. Add: (2x² + 3x - 1) + (x² - x + 2)
  6. Subtract: (x³ + x) - (2x³ - x²)
  7. Multiply: (x² + 1)(x - 2)
  8. Expand: (x + 3)²

If P(x) and G(x) are polynomials such that G(x) ≠ 0 and the degree of P(x) ≥ degree of G(x), then there exist unique polynomials Q(x) and R(x) such that:

P(x) = G(x) × Q(x) + R(x), where degree(R(x)) < degree(G(x))

This is called the Division Algorithm.

Solved Examples:

  • Divide x² + 3x + 2 by x + 1 → Quotient = x + 2, Remainder = 0
  • Divide x³ - 1 by x - 1 → Quotient = x² + x + 1, Remainder = 0
  • Divide 2x² + 5x + 3 by x + 1 → Quotient = 2x + 3, Remainder = 0
  • Divide x² - 1 by x - 1 → Quotient = x + 1, Remainder = 0
  • Divide x² + 4x + 3 by x + 3 → Quotient = x + 1, Remainder = 0

Exercise:

  1. Divide x² + 2x + 1 by x + 1
  2. Divide x³ - 3x² + 3x - 1 by x - 1
  3. Divide x² + x by x
  4. Divide 2x² - 3x - 2 by x - 2
  5. Divide x² - 4 by x - 2
  6. Divide 3x² + 7x + 2 by x + 2
  7. Divide x³ + x² - x - 1 by x + 1
  8. Divide x² - 2x + 1 by x - 1

Choose the correct answer for each of the following questions:

  1. The degree of the polynomial \(4x^3 + 2x^2 - x + 5\) is:
    • (a) 1
    • (b) 2
    • (c) 3
    • (d) 4
  2. Which of these is a quadratic polynomial?
    • (a) \(x^3 + 2x\)
    • (b) \(3x^2 + 5x + 1\)
    • (c) \(x + 1\)
    • (d) 2
  3. Which of the following is a binomial?
    • (a) \(x^2 + 2x + 1\)
    • (b) \(x^3\)
    • (c) \(x + 3\)
    • (d) 7
  4. The coefficient of \(x^2\) in \(5x^3 - 2x^2 + 3x\) is:
    • (a) 5
    • (b) -2
    • (c) 3
    • (d) 0
  5. Which one is a monomial?
    • (a) \(2x\)
    • (b) \(x + 1\)
    • (c) \(x^2 + x\)
    • (d) \(1 + x^2 + x^3\)
  6. The value of \((x+1)(x-1)\) is:
    • (a) \(x^2 + 1\)
    • (b) \(x^2 - 1\)
    • (c) \(x^2 - x + 1\)
    • (d) \(x^2 - 2x + 1\)
  7. Degree of \(x^5 + x^3 + 7\) is:
    • (a) 5
    • (b) 3
    • (c) 1
    • (d) 7
  8. \(x^2 + 4x + 4\) is a:
    • (a) Monomial
    • (b) Trinomial
    • (c) Binomial
    • (d) Constant
  9. The remainder when \(x^2 - 4\) is divided by \(x - 2\) is:
    • (a) 0
    • (b) 4
    • (c) -4
    • (d) 2
  10. Sum of \(3x^2 + 2x\) and \(x^2 + x + 1\) is:
    • (a) \(4x^2 + 3x + 1\)
    • (b) \(4x^2 + x + 1\)
    • (c) \(3x^2 + x + 1\)
    • (d) \(x^2 + 2x + 1\)
  11. The degree of a zero polynomial is:
    • (a) 0
    • (b) 1
    • (c) Undefined
    • (d) Not defined
  12. Which of the following is not a polynomial?
    • (a) \(x^2 - 3x + 1\)
    • (b) \(2x^{-1} + 1\)
    • (c) \(x^3 + 2x\)
    • (d) \(4x^4 + 1\)
  13. \((x + 2)^2\) simplifies to:
    • (a) \(x^2 + 4\)
    • (b) \(x^2 + 4x + 4\)
    • (c) \(x^2 - 4\)
    • (d) \(x^2 + 2x + 1\)
  14. \(x^2 - x - 6 = (x - 3)(x + 2)\) is an example of:
    • (a) Expansion
    • (b) Simplification
    • (c) Factorisation
    • (d) Substitution
  15. \((x + 1)^3\) expands to:
    • (a) \(x^3 + 3x^2 + 3x + 1\)
    • (b) \(x^3 + x + 1\)
    • (c) \(x^3 + 1\)
    • (d) \(x^2 + 2x + 1\)
Self Assessment Paper

PART 3: Self Assessment Paper

Section A: Multiple Choice Questions (1 mark each)

  1. Which of the following is a linear polynomial?
    1. x² + 2x + 3
    2. x + 5
    3. x³ + 1
    4. x²
  2. Degree of 7x⁵ + 2x³ + 5 is:
    1. 3
    2. 2
    3. 5
    4. 1
  3. 2x(x - 1) =
    1. 2x - 2
    2. 2x² - 2x
    3. 2x²
    4. 2x² + x
  4. A polynomial with one term is called:
    1. Binomial
    2. Monomial
    3. Trinomial
    4. Polynomial
  5. (x - 2)(x + 2) =
    1. x² + 4
    2. x² - 4
    3. x² + 2x - 4
    4. x² - 2x + 4
  6. Which of the following is not a polynomial?
    1. x² + 1
    2. x⁻¹ + 2
    3. x³ - x
    4. 5x + 3
  7. The constant term in 3x² + 5x - 7 is:
    1. 5
    2. -7
    3. 3
    4. 0
  8. x(x + 1)(x - 1) =
    1. x³ - x
    2. x³ + x
    3. x² - 1
    4. x³ - 1

Section B: Short Answer Questions (2 marks each)

  1. Define a polynomial and give one example.
  2. Find the degree and number of terms of 5x⁴ - 3x³ + x - 8.
  3. Simplify: (2x² + 3x - 4) + (x² - x + 1)
  4. Classify each polynomial as monomial, binomial, or trinomial: x, x+1, x²+x+1
  5. Divide x² + 5x + 6 by x + 3.
  6. Write the standard form of a cubic polynomial and give one example.

Section C: Long Answer Questions (4 marks each)

  1. Multiply: (x + 4)(x² - 2x + 3) and simplify.
  2. Divide x³ + 6x² + 11x + 6 by x + 1 using long division.
  3. Verify the identity (a + b)² = a² + 2ab + b² for a = x and b = 3.
  4. Explain with an example the process of polynomial multiplication with two binomials.

Section D: Case Study Based Question (5 marks)

Case Study:

A group of students decided to explore patterns in polynomial expressions through planting in their school garden. They planted flowers in square patches. One student expressed the area of a single flower patch as x² + 4x + 4, representing length and width both as (x + 2). The students then decided to place three such patches next to each other in a line and noticed the total area could also be expressed as a polynomial. They wanted to figure out how the polynomial changed with more patches and compared this area polynomial with others like x² + 2x + 1, etc.

  1. What is the polynomial expression for the area of one patch?
    1. x² + 4x + 4
    2. x² + 2x + 1
    3. x² + x + 1
    4. x² + 5x + 4
  2. What is the total area of three patches placed side by side?
    1. 3x² + 12x + 12
    2. x² + 12x + 12
    3. x² + 4x + 4
    4. 3x² + 4x + 4
  3. Which polynomial identity is used to derive area expression of one patch?
    1. (a + b)²
    2. (a - b)²
    3. (a + b)(a - b)
    4. a² - b²
  4. What is the degree of the polynomial x² + 4x + 4?
    1. 1
    2. 2
    3. 3
    4. 4
  5. Which of these is the square of a binomial?
    1. x² + 4x + 4
    2. x² + 4x + 1
    3. x² + x + 1
    4. x² - 4x + 4

Answer Key

Section A:

  1. b
  2. c
  3. b
  4. b
  5. b
  6. b
  7. b
  8. a

Section B to D: Only answer values (no steps required)

Section B:

  1. A polynomial is an expression consisting of variables and coefficients... Example: x² + 2x + 1
  2. Degree: 4, Number of terms: 4
  3. 3x² + 2x - 3
  4. Monomial: x, Binomial: x+1, Trinomial: x²+x+1
  5. Quotient: x + 2
  6. Standard cubic: ax³ + bx² + cx + d, e.g., x³ + 2x² - x + 4

Section C:

  1. x³ + 2x² + 7x + 12
  2. Quotient: x² + 5x + 6
  3. LHS: (x+3)² = x² + 6x + 9, RHS same, Verified
  4. Example: (x+2)(x+3) = x² + 5x + 6

Section D:

  1. a
  2. a
  3. a
  4. b
  5. a