CBSE Class 10 Mathematics Polynomials Notes Set 05

What is a zero of a polynomial?

A polynomial p(x) is an expression such as 2x² − 5x + 3 in which the powers of x are whole numbers. The degree is the highest power. A polynomial of degree 1 is linear, of degree 2 is quadratic and of degree 3 is cubic.

A real number k is a zero of p(x) if p(k) = 0. To test whether a number is a zero, substitute it for x and check that the value is zero. For example, x = −3 is not a solution of 2x² + 5x + 3 = 0, because 2(9) − 15 + 3 = 6, which is not zero.

Geometrical meaning of zeroes

The zeroes of p(x) are the x-coordinates of the points where the graph of y = p(x) meets the x-axis.

1. The graph of a linear polynomial ax + b is a straight line. It cuts the x-axis at exactly one point, so it has one zero.

2. The graph of a quadratic polynomial ax² + bx + c is a parabola. It opens upwards (like a U) if a > 0 and downwards (like an upside-down U) if a < 0.

3. A parabola can cut the x-axis at two points, touch it at one point, or not meet it at all. So a quadratic has two, one or no real zeroes.

4. In general, a polynomial of degree n has at most n zeroes.

Relation between zeroes and coefficients of a quadratic

If α and β are the zeroes of ax² + bx + c, where a ≠ 0, then:

1. Sum of zeroes: α + β = \( \dfrac{-b}{a} \) = −(coefficient of x) ÷ (coefficient of x²).

2. Product of zeroes: αβ = \( \dfrac{c}{a} \) = (constant term) ÷ (coefficient of x²).

3. A quadratic polynomial with zeroes α and β is k[x² − (α + β)x + αβ], where k is any non-zero real number. In words, it is x² − (sum of zeroes)x + (product of zeroes).

Worked examples

Example 1. Find the zeroes of x² − x − 6 and check the relation. Factorising, x² − 3x + 2x − 6 = (x − 3)(x + 2), so the zeroes are 3 and −2. The sum is 1, which equals −(−1)/1. The product is −6, which equals −6/1.

Example 2. Find a quadratic polynomial whose zeroes have sum 3 and product −2. The polynomial is x² − 3x − 2.

Example 3. If −4 is a zero of x² − x − (2k + 2), find k. Put x = −4: 16 + 4 − (2k + 2) = 0, so 2k = 18 and k = 9.

Example 4. If one zero of x² − 4x + 1 is 2 + √3, find the other. The sum of zeroes is 4, so the other zero is 4 − (2 + √3) = 2 − √3. The product (2 + √3)(2 − √3) = 1 matches the constant term.

Example 5. If α and β are the zeroes of 2x² + 7x + 5, find α + β + αβ. Here α + β = −7/2 and αβ = 5/2, so the value is −7/2 + 5/2 = −1.

Exam tip: always write the formulas with the coefficients a, b and c first, and then substitute. Check any final answer by testing the sum and product.

Quick Check Questions