CBSE Class 10 Mathematics Polynomials Notes

Polynomials is one of the most important chapters in Class 10 Mathematics. It builds on your knowledge of algebraic expressions and introduces the concepts of zeros of a polynomial, the relationship between zeros and coefficients, and the division algorithm for polynomials. Mastering this chapter is essential for scoring well in board examinations.

Key Topics Covered

•         What is a Polynomial? Types and Degree

•         Geometrical Meaning of Zeros of a Polynomial

•         Relationship Between Zeros and Coefficients

•         Division Algorithm for Polynomials

•         Quick Formula Summary

•         Board Exam Practice Questions

1.  What is a Polynomial?

A polynomial in one variable x is an algebraic expression of the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where aₙ ≠ 0 and all coefficients are real numbers. The highest power of x with a non-zero coefficient is called the degree of the polynomial.

Types of Polynomials by Degree

Polynomials are classified based on their degree:

Zeros of a Polynomial

A zero (or root) of a polynomial p(x) is a value of x for which p(x) = 0. A polynomial of degree n has at most n zeros. For example, for p(x) = x² – 5x + 6, the zeros are x = 2 and x = 3 since p(2) = 0 and p(3) = 0.

A polynomial of degree n can have at most n zeros.

2.  Geometrical Meaning of Zeros

The zeros of a polynomial correspond to the x-coordinates of the points where the graph of y = p(x) intersects the x-axis. This geometric interpretation helps you visualise how many zeros a polynomial has.

Graphs and Number of Zeros

3.  Relationship Between Zeros and Coefficients

This is the most important and most frequently tested part of the chapter. Learn these relationships thoroughly.

For a Linear Polynomial:  ax + b

If α is the zero of p(x) = ax + b, then:

α  =  –b / a  =  –(Constant term) / (Coefficient of x)

For a Quadratic Polynomial:  ax² + bx + c

If α and β are the two zeros of p(x) = ax² + bx + c, then:

Sum of zeros:  α + β  =  –b/a  =  –(Coefficient of x) / (Coefficient of x²)

Product of zeros:  αβ  =  c/a  =  (Constant term) / (Coefficient of x²)

Example:  For p(x) = x² – 5x + 6, find the sum and product of zeros.

a = 1,  b = –5,  c = 6

Sum of zeros:  α + β  =  –(–5)/1  =  5

Product of zeros:  αβ  =  6/1  =  6

Verification:  Zeros are 2 and 3.  Sum = 2 + 3 = 5 ✓    Product = 2 × 3 = 6 ✓

For a Cubic Polynomial:  ax³ + bx² + cx + d

If α, β, and γ are the three zeros of p(x) = ax³ + bx² + cx + d, then:

α + β + γ  =  –b/a

αβ + βγ + γα  =  c/a

αβγ  =  –d/a

To form a quadratic polynomial: x² – (Sum of zeros)x + (Product of zeros)

4.  Division Algorithm for Polynomials

Division Algorithm states: If p(x) and g(x) are any two polynomials with g(x) ≠ 0, then there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that:

p(x)  =  g(x) × q(x) + r(x)

where r(x) = 0  OR  degree of r(x) < degree of g(x). This is exactly analogous to the division of integers.

Steps to Apply Division Algorithm

1.       Step 1:  Arrange both dividend p(x) and divisor g(x) in descending powers of x.

2.       Step 2:  Divide the leading term of p(x) by the leading term of g(x) to get the first term of q(x).

3.       Step 3:  Multiply g(x) by this term, subtract from p(x), and bring down the next term.

4.       Step 4:  Repeat until the degree of the remainder is less than the degree of g(x).

5.       Step 5:  Verify:  p(x) = g(x) × q(x) + r(x)

Example:  Divide p(x) = x³ – 3x² + 5x – 3  by  g(x) = x² – 2

Quotient:   q(x)  =  x – 3

Remainder:  r(x)  =  7x – 9

Verify: (x²–2)(x–3) + (7x–9) = x³–3x²+5x–3  ✓

5.  Quick Formula Summary

All key formulas from this chapter that are frequently tested in CBSE board examinations:

6.  Board Exam Practice Questions

Commonly asked question types in CBSE Class 10 board examinations from this chapter:

1 Mark Questions

1.       If one zero of the polynomial p(x) = 5x² + 13x + k is the reciprocal of the other, find k.

2.       Find the sum and product of zeros of the polynomial x² – 7.

3.       If α and β are zeros of 2x² – 5x + 7, find α + β.

4.       How many zeros does the polynomial (x – 2)² – 4 have?

3 Mark Questions

1.       Find a quadratic polynomial whose zeros are 3 + √2 and 3 – √2.

2.       If the zeros of p(x) = x² + px + q are double in value to the zeros of 2x² – 5x – 3, find p and q.

3.       Verify that 2, –1, and –3 are zeros of the cubic polynomial p(x) = x³ + 2x² – 5x – 6 and verify the relationship between zeros and coefficients.

5 Mark Questions

1.       Divide p(x) = x³ – 3x² + 5x – 3 by g(x) = x² – 2 and verify using the Division Algorithm.

2.       Given that x – √5 is a factor of the cubic polynomial x³ – 3√5x² + 13x – 3√5, find all zeros of the polynomial.

3.       If two zeros of p(x) = x⁴ – 6x³ – 26x² + 138x – 35 are 2 ± √3, find the other two zeros.