Polynomials — Study Notes

1. Introduction

Polynomials are among the most important algebraic expressions studied in mathematics. A polynomial in the variable x is an expression of the form a_n x^n + a_(n-1) x^(n-1) + ... + a_1 x + a_0, where n is a non-negative integer and the coefficients a_i are real numbers. This chapter builds on the Class 9 foundation of polynomials and introduces the crucial relationship between the zeroes of a polynomial and its coefficients.

The chapter systematically covers the zeroes of linear, quadratic, and cubic polynomials, the graphical meaning of zeroes, and the factor theorem. Students learn how to verify whether a number is a zero of a polynomial, how to find a quadratic polynomial when its zeroes are given, and how to divide one polynomial by another using the division algorithm for polynomials.

Understanding polynomials is indispensable for higher mathematics. Every subsequent algebraic chapter, including quadratic equations, arithmetic progressions, and coordinate geometry, relies heavily on polynomial concepts. The geometric and algebraic interpretations developed here also prepare students for calculus topics like derivatives in higher classes.

2. Types of Polynomials

Based on Number of Terms

• Monomial: one term, e.g., 5x, 7.
• Binomial: two terms, e.g., 3x + 2.
• Trinomial: three terms, e.g., x^2 + 2x + 1.

Based on Degree

The degree of a polynomial is the highest power of the variable in it. - Linear polynomial: degree 1, form ax + b, a is not 0. Example: 2x + 3. - Quadratic polynomial: degree 2, form ax^2 + bx + c, a is not 0. Example: x^2 - 3x + 2. - Cubic polynomial: degree 3, form ax^3 + bx^2 + cx + d, a is not 0. Example: x^3 - 4x. - A constant polynomial like 7 has degree 0. The zero polynomial 0 has no defined degree.

A polynomial of one term is also a monomial; every polynomial can have a degree that tells us the maximum number of zeroes it can have.

3. Zeroes of a Polynomial

A real number k is said to be a zero of the polynomial p(x) if p(k) = 0. Graphically, the zero of a polynomial is the x-coordinate of the point where the graph of y = p(x) intersects the x-axis.

Number of Zeroes

• A linear polynomial (degree 1) has exactly one zero.
• A quadratic polynomial (degree 2) has at most two zeroes.
• A cubic polynomial (degree 3) has at most three zeroes.
• In general, a polynomial of degree n has at most n real zeroes.

Example: For p(x) = x^2 - 3x + 2, we have p(1) = 1 - 3 + 2 = 0 and p(2) = 4 - 6 + 2 = 0. Hence 1 and 2 are zeroes. Note that the graph of a quadratic crosses the x-axis at these two points.

Geometrical Meaning

The number of zeroes of a polynomial equals the number of times its graph cuts or touches the x-axis. For a quadratic polynomial, the parabola may cut the axis at two points (two zeroes), touch it at one point (one zero), or not touch it at all (no real zeroes).

4. Relationship Between Zeroes and Coefficients

Quadratic Polynomial ax^2 + bx + c

If alpha and beta are the zeroes, then: 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)

Cubic Polynomial ax^3 + bx^2 + cx + d

If alpha, beta, and gamma are the zeroes, then: alpha + beta + gamma = -b/a alpha x beta + beta x gamma + gamma x alpha = c/a alpha x beta x gamma = -d/a

Example: For p(x) = x^2 - 5x + 6, zeroes are 2 and 3. Sum = 2 + 3 = 5 = -(-5)/1. Product = 2 x 3 = 6 = 6/1. Verified.

5. Forming a Quadratic Polynomial from Its Zeroes

If alpha and beta are the zeroes, the quadratic polynomial is: k[x^2 - (alpha + beta)x + alpha x beta]

Where k is a non-zero constant. If we set k = 1, we get the polynomial with sum of zeroes S and product P as: x^2 - Sx + P

Example: Zeroes 3 and -5. S = 3 + (-5) = -2, P = 3 x (-5) = -15. Polynomial = x^2 - (-2)x + (-15) = x^2 + 2x - 15.

6. Division Algorithm for Polynomials

If p(x) and g(x) are two polynomials with g(x) not equal to zero, then there exist 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). Here p(x) is the dividend, g(x) the divisor, q(x) the quotient, and r(x) the remainder.

Steps for Long Division

1. Arrange the dividend and divisor in descending order of degrees.
2. Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
3. Multiply the divisor by this term, subtract from the dividend, and bring down the next term.
4. Repeat until the remainder is of lower degree than the divisor.

This algorithm verifies whether a given factor is a divisor, and it is used to find all zeroes of a cubic polynomial when one or two zeroes are known.

7. Factor Theorem

If p(x) is a polynomial of degree greater than or equal to 1 and a is any real number, then: - x - a is a factor of p(x) if and only if p(a) = 0.

Example: For p(x) = x^3 - 6x^2 + 11x - 6, p(1) = 1 - 6 + 11 - 6 = 0. Hence x - 1 is a factor. Dividing, we get (x - 1)(x^2 - 5x + 6) = (x - 1)(x - 2)(x - 3), giving zeroes 1, 2, 3.

The factor theorem is a special case of the remainder theorem where the remainder is zero.

Quick Revision Tables

Table 1: Polynomials by Degree

Table 2: Zero-Coefficient Relationships

Mind Map

Important Diagrams (SVG)

Diagram 1: Graph of a Quadratic Polynomial and its Zeroes

Diagram 2: Zero-Coefficient Relationship Visual

8. Solved Example Approach

A common board question asks us to find a quadratic polynomial whose zeroes are given. The procedure is to compute the sum and the product of the zeroes, then substitute them into the form x^2 - (sum)x + (product). For zeroes 2 and -3, the sum is 2 + (-3) = -1 and the product is 2 x (-3) = -6, so the polynomial is x^2 - (-1)x + (-6) = x^2 + x - 6. We can verify the answer by checking that p(2) = 4 + 2 - 6 = 0 and p(-3) = 9 - 3 - 6 = 0, so both given numbers are indeed zeroes.

A second type of question verifies the relationship between zeroes and coefficients. For the polynomial p(x) = 2x^2 - 8x + 6, we first factor it as 2(x - 1)(x - 3), so the zeroes are 1 and 3. The sum of the zeroes is 1 + 3 = 4, and -b/a equals -(-8)/2 = 4, which matches. The product of the zeroes is 1 x 3 = 3, and c/a equals 6/2 = 3, which also matches. Performing this verification in the examination shows the examiner that both parts of the relationship are understood, and it also confirms that the factorisation was correct.

For a cubic polynomial, the factor theorem is the standard entry point. To find the zeroes of x^3 - 6x^2 + 11x - 6, we test small values and find p(1) = 0, so x - 1 is a factor. Dividing the cubic by x - 1 gives x^2 - 5x + 6, which factors as (x - 2)(x - 3). The zeroes are therefore 1, 2, and 3, and we can check the three relationships between zeroes and coefficients, such as the sum being -b/a = 6. This pattern of factor, divide, and factor again solves almost every cubic problem in the syllabus, and it shows how the factor theorem and the division algorithm work together as one complete method.

Common Mistakes

1. Confusing the coefficient of x with the constant term. In sum of zeroes = -b/a, the b is the coefficient of x, and the sign is negative.
2. Writing the polynomial from zeroes as x^2 + Sx + P instead of x^2 - Sx + P. The sum S enters with a negative sign.
3. Forgetting that the degree of the zero polynomial is not defined, and that a constant polynomial has degree zero.
4. Stating that a quadratic polynomial always has two real zeroes. A quadratic may have one, two, or zero real zeroes depending on the graph.
5. In the division algorithm, forgetting the condition degree of r(x) < degree of g(x), or writing the remainder of degree greater than or equal to the divisor.
6. Using the factor theorem without checking: claiming x - a is a factor even when p(a) is not zero.
7. Mixing up the pairwise sum formula for cubic polynomials: alpha x beta + beta x gamma + gamma x alpha = c/a, not the simple sum.

Exam Tips

1. Memorise the sign conventions: sum = -b/a (quadratic) and -b/a for the first relation in cubic; product = c/a (quadratic) and -d/a (cubic).
2. To verify zeroes, substitute the value into the polynomial and show the result is zero with proper steps.
3. When finding a polynomial from given zeroes, compute S and P first, then write x^2 - Sx + P, and mention k for the general form.
4. In division algorithm questions, always write p(x) = g(x)q(x) + r(x) explicitly and state that r(x) = 0 or deg r(x) < deg g(x).
5. For cubic polynomials, use the factor theorem to find one factor, then divide to get a quadratic, and solve it.
6. Practise graphs: be able to sketch parabolas (upward if a > 0, downward if a < 0) and read the number of zeroes from the graph.
7. Attempt previous-year board questions on "find a quadratic polynomial whose sum and product of zeroes are given" as it is a guaranteed exam favourite.

Conclusion

Polynomials form a bridge between basic algebra and higher analysis. The relationship between zeroes and coefficients, the division algorithm, and the factor theorem together provide a complete toolkit for solving polynomial problems systematically. This chapter not only appears directly in board exams but also underpins quadratic equations, factorisation techniques, and graph interpretation used across the syllabus. A thorough grasp of zeroes, their sum and product, and polynomial division will give students a strong algebraic foundation for all later topics in mathematics.