Mathematics | Class 10 Mathematics Notes

2.1 Introduction

This section revisits the concept of polynomials in one variable, a topic introduced in Class IX. A polynomial p(x) in variable x is an algebraic expression involving terms with non-negative integer powers of x and real coefficients. The degree of a polynomial is the highest power of the variable present in it. For example, 4x + 2 is a polynomial of degree 1, 2y² - 3y + 4 is degree 2, 5x³ - 4x² + x - √2 is degree 3, and 7u⁶ - (3/2)u³ + 4u² + u - 8 is degree 6. Expressions such as 1/(x - 1), √x + 2, or 1/(x² + 2x + 3) are not polynomials because they involve negative or fractional powers or variables in denominators. Polynomials are classified by their degree: degree 1 polynomials are linear, degree 2 are quadratic, and degree 3 are cubic. Linear polynomials have the form ax + b where a ≠ 0; quadratic polynomials have the form ax² + bx + c with a ≠ 0; cubic polynomials have the form ax³ + bx² + cx + d with a ≠ 0. The section also introduces the concept of the value of a polynomial at a given point k, denoted p(k), obtained by substituting x = k in p(x). A zero of a polynomial is a value k for which p(k) = 0. For example, the polynomial x² - 3x - 4 has zeros at x = -1 and x = 4 because p(-1) = 0 and p(4) = 0. The zero of a linear polynomial ax + b is -b/a, showing a direct relation between zeros and coefficients. The section poses the question of whether such relationships exist for polynomials of higher degrees, which will be explored further. Finally, it introduces the division algorithm for polynomials as a topic to be studied.

🔗 Connection: Leads to the next section which explores the geometrical meaning of zeros of polynomials by studying their graphs.