Introduction
This chapter explores the relationship between the zeroes of a polynomial and its coefficients. You will learn how to find the zeroes of quadratic polynomials graphically and algebraically, and understand the relationship between the zeroes and the coefficients of quadratic polynomials. The chapter builds on your understanding of linear and quadratic expressions from earlier classes.
Geometrical Meaning of Zeroes
The zeroes of a polynomial p(x) are the x-coordinates of the points where the graph of y = p(x) intersects the x-axis. For a linear polynomial ax + b, the graph is a straight line that intersects the x-axis at exactly one point, giving exactly one zero. For a quadratic polynomial ax^2 + bx + c, the graph is a parabola that can intersect the x-axis at 0, 1, or 2 points, giving 0, 1, or 2 zeroes respectively.
Key Points
- •A linear polynomial has exactly one zero
- •A quadratic polynomial has at most two zeroes
- •A polynomial of degree n has at most n zeroes
- •If the parabola does not touch the x-axis, the quadratic has no real zeroes
- •The parabola opens upward if a > 0 and downward if a < 0
Worked Example
For p(x) = x^2 - 3x + 2: Setting p(x) = 0: x^2 - 3x + 2 = 0 (x - 1)(x - 2) = 0 Zeroes are x = 1 and x = 2. The parabola crosses the x-axis at (1, 0) and (2, 0).
Watch Out
When asked to find the number of zeroes from a graph, count the number of times the curve crosses the x-axis.
Relationship Between Zeroes and Coefficients
For a quadratic polynomial ax^2 + bx + c with zeroes alpha and beta, there is a direct relationship between the zeroes and the coefficients. The sum of zeroes equals -b/a and the product of zeroes equals c/a. These relationships are extremely useful for forming polynomials when zeroes are given, and for finding one zero when the other is known.
Key Points
- •Sum of zeroes (alpha + beta) = -b/a = -(coefficient of x)/(coefficient of x^2)
- •Product of zeroes (alpha x beta) = c/a = (constant term)/(coefficient of x^2)
- •A quadratic polynomial with zeroes alpha and beta is k[x^2 - (alpha+beta)x + alpha.beta]
- •k is any non-zero constant (usually taken as 1)
- •These relationships come from comparing ax^2 + bx + c with a(x - alpha)(x - beta)
Worked Example
Find a quadratic polynomial whose zeroes are 3 and -2. Sum of zeroes = 3 + (-2) = 1 Product of zeroes = 3 x (-2) = -6 Polynomial = x^2 - (sum)x + (product) = x^2 - x - 6 Verification: x^2 - x - 6 = (x - 3)(x + 2). Zeroes are 3 and -2. Correct!
Watch Out
Always verify your answer by substituting the zeroes back into the polynomial.
Finding Zeroes of a Quadratic Polynomial
The zeroes of a quadratic polynomial ax^2 + bx + c can be found by factorising the polynomial or by using the quadratic formula. For factorisation, we split the middle term bx into two parts whose product equals ac. The quadratic formula gives zeroes as x = [-b +/- sqrt(b^2 - 4ac)] / 2a. The discriminant D = b^2 - 4ac determines the nature of zeroes: if D > 0, two distinct real zeroes; if D = 0, two equal real zeroes; if D < 0, no real zeroes.
Key Points
- •Splitting the middle term: find two numbers that add to b and multiply to ac
- •Quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / 2a
- •D > 0: two distinct real zeroes
- •D = 0: two equal real zeroes (repeated root)
- •D < 0: no real zeroes
Worked Example
Find the zeroes of 6x^2 - 7x - 3. a = 6, b = -7, c = -3 Product ac = 6 x (-3) = -18 Find two numbers that add to -7 and multiply to -18: -9 and 2 6x^2 - 9x + 2x - 3 = 3x(2x - 3) + 1(2x - 3) = (3x + 1)(2x - 3) Zeroes: x = -1/3 and x = 3/2 Verification: Sum = -1/3 + 3/2 = 7/6 = -b/a. Product = -1/3 x 3/2 = -1/2 = c/a. Verified!
Division Algorithm for Polynomials (Overview)
If p(x) and g(x) are any two polynomials with g(x) not equal to zero, then we can find polynomials q(x) and r(x) such that p(x) = g(x) x q(x) + r(x), where r(x) = 0 or degree of r(x) < degree of g(x). This is similar to the division algorithm for integers. Note: Detailed applications of the division algorithm are not in the current CBSE syllabus, but the concept is important.
Key Points
- •p(x) = g(x) x q(x) + r(x) where degree of r(x) < degree of g(x)
- •If r(x) = 0, then g(x) is a factor of p(x)
- •This is analogous to Dividend = Divisor x Quotient + Remainder for integers
- •The concept is useful for verifying factors of polynomials
Watch Out
While the division algorithm proof is deleted from CBSE 2025-26, understanding the concept helps in verifying factorisation.
Quick Summary
- ✓Zeroes of a polynomial are where its graph crosses the x-axis
- ✓A polynomial of degree n has at most n zeroes
- ✓For ax^2 + bx + c: Sum of zeroes = -b/a
- ✓For ax^2 + bx + c: Product of zeroes = c/a
- ✓Quadratic with zeroes alpha, beta: x^2 - (alpha+beta)x + alpha.beta
- ✓Zeroes found by: splitting middle term, or quadratic formula
- ✓Discriminant D = b^2 - 4ac determines nature of zeroes
Key Formulas
Sum of zeroes = -b/a
Product of zeroes = c/a
Quadratic polynomial from zeroes: k[x^2 - (sum)x + (product)]
Quadratic formula: x = [-b +/- sqrt(b^2 - 4ac)] / 2a
Discriminant D = b^2 - 4ac
Ready to practice?
Test your understanding with questions