Polynomials

Polynomials Notes: Formulas, Theorems & Examples (Class 10 Maths)

Ch 2 : POLYNOMIALS

These notes are designed to help you understand Polynomials easily, even if you’re learning on your own. They cover everything in the CBSE/NCERT Class 10 syllabus and will give you a quick but detailed overview of the chapter for effective revision.

CONTENT OF LESSON:

  • Variables
  • Constants
  • Algebraic Expressions
  • Polynomials
  • Degree of a Polynomial
  • Types of Polynomials
  • Geometrical Representation of Polynomials
  • Zeroes of a Polynomial
  • Relationship between Zeroes and Coefficients of Quadratic and Cubic Polynomials
  • Factorization of Quadratic Polynomials
  • Division Algorithm for Polynomials

 

CHAPTER CONCEPTS:

1. VARIABLES:

  • DEFINITION: Variables are symbols (usually letters like x, y, z) that represent values that can change or vary. In polynomial expressions, they are the unknowns we work with.
  • EXAMPLES: In the expression 2x + 3, ‘x’ is the variable. In xy – 5, ‘x’ and ‘y’ are variables.

2. CONSTANTS:

  • DEFINITION: Constants are fixed numerical values that do not change. They are the numbers in polynomial expressions.
  • EXAMPLES: In the expression 2x + 3, ‘3’ and ‘2’ are constants. In xy – 5, ‘-5’ is a constant. Other examples: 5, -7, 1/2, √2, π are all constants.

3. ALGEBRAIC EXPRESSIONS:

  • DEFINITION: An algebraic expression is an expression made up of variables, constants, and mathematical operations (+, -, ×, ÷, exponents, roots, etc.). They are the foundation for polynomials.
  • KEY POINT: Algebraic expressions become equations when they are set equal to something (e.g., 3x + 2 = 5 is an equation, while 3x + 2 is an algebraic expression).
  • EXAMPLES:
    • 3x²y + 4xy + 5x + 6
    • √x + 7
    • (x + 1) / (x – 2)
    • 5x⁻¹ + 2
  • TERMS: An algebraic expression is a sum of terms, which are considered to be building blocks for expressions. Terms are separated by ‘+’ or ‘-‘ signs.
    • Constant terms: (e.g., 5, -3)
    • Variable terms: (e.g., 2x, -4xy, 3x²)
    • Like terms: (terms with the same variables raised to the same powers, e.g., 2x and 5x, 3x²y and -x²y)

4. POLYNOMIALS:

  • DEFINITION: A polynomial is a special type of algebraic expression in which the exponent on every variable is a whole number (0, 1, 2, 3,…).
  • WHOLE NUMBERS: Remember, whole numbers are non-negative integers: 0, 1, 2, 3, …
  • KEY DIFFERENCE from Algebraic Expressions: In general algebraic expressions, exponents can be any rational number. In polynomials, exponents are restricted to whole numbers.
  • EXAMPLES OF POLYNOMIALS:
    • 5x³ + 3x + 1 (Exponents are 3, 1, and 0 for the constant term)
    • x² – 7x + 10 (Exponents are 2, 1, and 0)
    • 7 (Constant polynomial, can be written as 7x⁰, exponent is 0)
    • -2y⁴ + y² – 5y + 8
  • EXAMPLES OF NON-POLYNOMIALS (Algebraic Expressions but NOT Polynomials):
    • 2x + 3√x because √x = x^(1/2), and 1/2 is NOT a whole number.
    • x⁻¹ + 2x because x⁻¹ = 1/x, and -1 is NOT a whole number.
    • 3 + 2/x because 2/x = 2x⁻¹, and -1 is NOT a whole number.
    • x² + x^(1.5) – 1 because 1.5 is NOT a whole number.

5. DEGREE OF A POLYNOMIAL:

  • DEFINITION: For a polynomial in one variable, the degree of the polynomial is the highest exponent on the variable in the polynomial.
  • EXAMPLES:
    • x² + 2x + 3: Degree is 2 (highest power of x is 2).
    • 5x³ – 4x² + x – 7: Degree is 3 (highest power of x is 3).
    • 2x – 9: Degree is 1 (highest power of x is 1).
    • 6: Degree is 0 (constant polynomial, can be written as 6x⁰).
    • 0: Degree is not defined (zero polynomial).

6. TYPES OF POLYNOMIALS:

Polynomials can be classified based on two criteria:

(a) Based on the NUMBER OF TERMS:

  • MONOMIAL: A polynomial with just one term.
    • Examples: 2x, 6x², 9xy, -7x⁵, 10, ½y³, -√3z⁷
    • In essence: Single term polynomials.
  • BINOMIAL: A polynomial with two terms.
    • Examples: 4x² + x, 5x + 4, x¹⁰ – 1, 3y + 7y², √2z – 5, a + b
    • In essence: Two terms joined by ‘+’ or ‘-‘.
  • TRINOMIAL: A polynomial with three terms.
    • Examples: x² + 3x + 4, x² + 2xy + y², p³ – 4p + 9, x² + x + 1, a² + b² + c²
    • In essence: Three terms joined by ‘+’ or ‘-‘.

(b) Based on the DEGREE of the polynomial:

  • LINEAR POLYNOMIAL: A polynomial whose degree is one.
    • General form: ax + b (where a ≠ 0)
    • Examples: 2x + 1, -x + 5, 7y, ½z – 3, x – √7
    • In essence: Highest power of variable is 1.
  • QUADRATIC POLYNOMIAL: A polynomial of degree two.
    • General form: ax² + bx + c (where a ≠ 0)
    • Examples: 3x² + 8x + 5, -2x² + 3x – 1, y² + 9, √5z², x² – 4x + 4
    • In essence: Highest power of variable is 2.
  • CUBIC POLYNOMIAL: A polynomial of degree three.
    • General form: ax³ + bx² + cx + d (where a ≠ 0)
    • Examples: 2x³ + 5x² + 9x + 15, x³ – 1, 4y³ + 2y² – y + 6, -z³ + 8z, x³ + 3x² + 3x + 1
    • In essence: Highest power of variable is 3.
  • CONSTANT POLYNOMIAL: A polynomial of degree zero (a non-zero constant).
    • General form: c (where c ≠ 0)
    • Examples: 5, -3, √2, π, -1/7
    • In essence: Just a number (not zero).
  • ZERO POLYNOMIAL: The polynomial 0. Its degree is not defined.

7. GRAPHICAL REPRESENTATIONS OF POLYNOMIALS:

  • REPRESENTING EQUATIONS ON A GRAPH: Any equation can be represented as a graph on the Cartesian plane (x-y plane). Each point on the graph represents the (x, y) coordinates that satisfy the equation.
  • LINEAR POLYNOMIAL (y = ax + b):
  • The graph of a linear polynomial is always a straight line.
  • It cuts the x-axis at exactly one point (unless it’s a horizontal line not on the x-axis, or a vertical line which is not a function of x in this form).
  • WHY one x-intercept? A linear polynomial has degree 1, so at most one zero. This zero is the x-intercept.
  • The line can slope upwards (positive ‘a’), downwards (negative ‘a’), or be horizontal (a=0, constant function).

line_crossing_xaxis

  • QUADRATIC POLYNOMIAL (y = ax² + bx + c):
  • The graph of a quadratic polynomial is always a parabola.
  • It is a U-shaped curve.
  • It opens upwards if ‘a’ > 0 (positive coefficient of x²) – like a smile.
  • It opens downwards if ‘a’ < 0 (negative coefficient of x²) – like a frown.
  • It can cut the x-axis at 0, 1, or 2 points.
  • WHY 0, 1, or 2 x-intercepts? A quadratic polynomial has degree 2, so at most two zeros. These zeros are the x-intercepts.
    • Two distinct zeros: Parabola intersects x-axis at two points.
    • One zero (repeated zero): Parabola touches x-axis at one point (vertex on x-axis).
    • Zero zeros (no real zeros): Parabola does not intersect x-axis.

three_parabolas

8. ZEROES OF A POLYNOMIAL:

  • DEFINITION: A zero of a polynomial p(x) is the value of x for which the value of the polynomial becomes zero, i.e., p(x) = 0. If k is a zero of p(x), then p(k) = 0.
  • GEOMETRICAL MEANING: Geometrically, the zeroes of a polynomial are the x-coordinates of the points where the graph of the polynomial intersects the x-axis. These are also called x-intercepts.
  • EXAMPLE: For the polynomial p(x) = x² – 3x + 2:
  • If we substitute x = 1, we get p(1) = (1)² – 3(1) + 2 = 1 – 3 + 2 = 0. So, 1 is a zero of p(x).
  • If we substitute x = 2, we get p(2) = (2)² – 3(2) + 2 = 4 – 6 + 2 = 0. So, 2 is also a zero of p(x).
  • NUMBER OF ZEROES: In general, a polynomial of degree n has at most n zeros.
  • WHY “at most”? A polynomial of degree ‘n’ can have up to ‘n’ zeros. It might have fewer distinct real zeros if some are repeated or if some zeros are complex numbers (though in Class 10, you mainly focus on real zeros).
  • Linear polynomial (degree 1): Has at most 1 zero.
  • Quadratic polynomial (degree 2): Has at most 2 zeros.
  • Cubic polynomial (degree 3): Has at most 3 zeros.

9. FACTORISATION OF QUADRATIC POLYNOMIALS (Splitting the Middle Term):

  • METHOD: Quadratic polynomials (of the form ax² + bx + c) can often be factorized by splitting the middle term (bx).
  • STEPS FOR SPLITTING THE MIDDLE TERM:
  1. Standard Form: Ensure the quadratic polynomial is in the standard form ax² + bx + c.
  2. Product ‘ac’: Find the product of the coefficient of x² (a) and the constant term (c).
  3. Find two numbers: Find two numbers whose product is ‘ac’ and whose sum is the coefficient of x (b).
  4. Split the middle term: Rewrite the middle term (bx) as the sum of the two numbers found in step 3, each multiplied by ‘x’.
  5. Factor by grouping: Group the first two terms and the last two terms and factor out the common factors from each group.
  6. Common binomial factor: You should now have a common binomial factor in both groups. Factor out this common binomial factor.
  • EXAMPLE: Factorize 2x² – 5x + 3
  1. Standard form: It is already in standard form. a=2, b=-5, c=3.
  2. Product ‘ac’: ac = 2 * 3 = 6.
  3. Find two numbers: We need two numbers that multiply to 6 and add up to -5. These numbers are -2 and -3 (because (-2) * (-3) = 6 and (-2) + (-3) = -5).
  4. Split the middle term: Rewrite -5x as -2x – 3x. So, 2x² – 5x + 3 = 2x² – 2x – 3x + 3.
  5. Factor by grouping:
    • Group 1st two terms: (2x² – 2x) = 2x(x – 1)
    • Group last two terms: (-3x + 3) = -3(x – 1)
    • So, 2x² – 2x – 3x + 3 = 2x(x – 1) – 3(x – 1)
  6. Common binomial factor: The common factor is (x – 1). Factor it out: (x – 1)(2x – 3).

Therefore, 2x² – 5x + 3 = (x – 1)(2x – 3).

10. RELATIONSHIP BETWEEN ZEROES AND COEFFICIENTS OF A POLYNOMIAL:

(a) For QUADRATIC POLYNOMIAL: ax² + bx + c

  • Let α and β (alpha and beta) be the two zeroes of the quadratic polynomial ax² + bx + c. Then, the following relationships hold:
  • SUM OF ZEROES (α + β):
    • α + β = -b/a
    • In words: Sum of zeroes = -(coefficient of x) / (coefficient of x²)
  • PRODUCT OF ZEROES (αβ):
    • αβ = c/a
    • In words: Product of zeroes = (constant term) / (coefficient of x²)
  • UTILITY: These relationships are useful for:
    • Verifying your zeroes: Check if calculated zeroes are correct.
    • Finding zeroes if one is known.
    • Constructing a quadratic polynomial if sum and product of zeroes are given.

(b) For CUBIC POLYNOMIAL: ax³ + bx² + cx + d

  • Let α, β, and γ (alpha, beta, and gamma) be the three zeroes of the cubic polynomial ax³ + bx² + cx + d. Then, the following relationships hold:
  • SUM OF ZEROES (α + β + γ):
    • α + β + γ = -b/a
    • In words: Sum of zeroes = – (coefficient of x²) / (coefficient of x³)
  • SUM OF PRODUCTS OF ZEROES TAKEN TWO AT A TIME (αβ + βγ + γα):
    • αβ + βγ + γα = c/a
    • In words: Sum of products of zeroes (taken two at a time) = (coefficient of x) / (coefficient of x³)
  • PRODUCT OF ZEROES (αβγ):
    • αβγ = -d/a
    • In words: Product of zeroes = – (constant term) / (coefficient of x³)
  • UTILITY: Similar to quadratic polynomials, these relationships are useful for verification, finding zeroes, and constructing cubic polynomials (though constructions are less common in basic problems).

11. DIVISION ALGORITHM FOR POLYNOMIALS (Statement and Simple Problems):

  • STATEMENT: If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that:

p(x) = g(x) * q(x) + r(x)

where either r(x) = 0 or degree of r(x) < degree of g(x).

  • p(x) is the dividend (the polynomial being divided).
  • g(x) is the divisor (the polynomial we are dividing by).
  • q(x) is the quotient (the result of the division).
  • r(x) is the remainder (what’s left over after division).
  • ANALOGY TO NUMBERS: This is similar to the division algorithm for integers: Dividend = Divisor × Quotient + Remainder.
  • SIMPLE PROBLEMS: You will be asked to divide one polynomial p(x) by another polynomial g(x) and find the quotient q(x) and remainder r(x). You may also be asked to verify the division algorithm.