Polynomials — Study Notes

1. Introduction

Algebra is the branch of mathematics in which we use letters to represent numbers and study the rules of combining them. A very important class of algebraic expressions is the polynomial, which appears everywhere in mathematics: in geometry, in physics, in the graphs we draw and in the equations we solve. In this chapter we will study what a polynomial is, how to classify polynomials by their number of terms and by their degree, and how to evaluate them at given values of the variable.

The most exciting part of this chapter is the factorisation of polynomials. We will learn two powerful theorems: the remainder theorem and the factor theorem, which tell us how a polynomial behaves when we divide it by a linear polynomial. Along with these theorems we will study a set of standard algebraic identities that allow us to expand and factorise expressions quickly without lengthy multiplication. A strong command over polynomials is absolutely necessary for Class 10, where polynomials, quadratic equations and arithmetic progression problems build directly on the ideas introduced here.

2. What is a Polynomial?

A polynomial in one variable x is an algebraic expression of the form a(0) + a(1)x + a(2)x^2 + ... + a(n)x^n, where a(0), a(1), ..., a(n) are constants (numbers) and the powers of x are whole numbers. The constants are called the coefficients of the polynomial. For example, 3x^2 + 5x - 7 is a polynomial in x.

• The highest power of the variable in a polynomial is called the degree of the polynomial. In 3x^2 + 5x - 7, the degree is 2.
• A polynomial of degree 1 is called a linear polynomial, for example 2x + 3.
• A polynomial of degree 2 is called a quadratic polynomial, for example x^2 - 4x + 4.
• A polynomial of degree 3 is called a cubic polynomial, for example x^3 + 1.

Terms and Coefficients

The separate expressions added in a polynomial are called its terms. In 4x^3 + 2x^2 - x + 9, the terms are 4x^3, 2x^2, -x and 9. The constant multiplying the variable in a term is the coefficient: the coefficient of x^3 is 4, of x^2 is 2, and of x is -1. The term 9, which has no variable, is called the constant term.

Number of Terms

A polynomial having only one term is called a monomial (for example, 5x^2). A polynomial having two terms is called a binomial (for example, 2x + 1), and a polynomial having three terms is called a trinomial (for example, x^2 + x + 1). Note that expressions like sqrt x and 1/x are NOT polynomials, because the power of x is not a whole number.

3. Zeroes of a Polynomial

Consider a polynomial p(x). If we substitute a number a for x, we obtain the value p(a), which is called the value of the polynomial at x = a. For example, if p(x) = x^2 - 5x + 6, then p(2) = 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0.

A real number a is called a zero (or root) of the polynomial p(x) if p(a) = 0. In the example above, x = 2 is a zero of x^2 - 5x + 6 because p(2) = 0. Similarly, x = 3 is also a zero because p(3) = 9 - 15 + 6 = 0.

Important facts about zeroes: - Every linear polynomial ax + b (with a not equal to 0) has exactly one zero, given by x = -b/a. - A polynomial of degree n can have at most n distinct real zeroes. - The zero of a polynomial is the x-coordinate of the point where its graph intersects the x-axis. - Zero is a zero of p(x) if and only if the constant term of p(x) is zero, since p(0) = constant term.

4. Remainder Theorem

The remainder theorem gives a shortcut for dividing a polynomial by a linear polynomial of the form (x - a). It states:

If a polynomial p(x) is divided by (x - a), then the remainder is p(a).

That is, we do not need to perform the full long division to find the remainder; we simply evaluate the polynomial at x = a. For example, let p(x) = x^3 - 3x^2 + 4x + 2. The remainder when p(x) is divided by (x - 1) is p(1) = 1 - 3 + 4 + 2 = 4. If we were to divide by (x + 2), which is the same as (x - (-2)), the remainder would be p(-2) = (-8) - 12 - 8 + 2 = -26.

This theorem is extremely useful because dividing polynomials by long division is lengthy and error-prone, while substituting a number is quick and accurate.

5. Factor Theorem

The factor theorem is a special case of the remainder theorem. It states:

If p(a) = 0, then (x - a) is a factor of the polynomial p(x). Conversely, if (x - a) is a factor of p(x), then p(a) = 0.

In other words, a linear polynomial (x - a) divides p(x) completely (with zero remainder) exactly when a is a zero of the polynomial. For example, since p(2) = 0 for p(x) = x^2 - 5x + 6, we know that (x - 2) is a factor. Indeed, x^2 - 5x + 6 = (x - 2)(x - 3).

Factorising a Cubic Polynomial

To factorise a cubic polynomial, we first find one zero by trial (checking factors of the constant term), then divide by the corresponding factor using synthetic division or long division, and finally factorise the resulting quadratic. For example, for p(x) = x^3 - 6x^2 + 11x - 6, we try x = 1: p(1) = 1 - 6 + 11 - 6 = 0, so (x - 1) is a factor. Dividing, we get x^2 - 5x + 6 = (x - 2)(x - 3). Hence p(x) = (x - 1)(x - 2)(x - 3).

6. Algebraic Identities

An identity is an equality that is true for every value of the variable. The following identities must be memorised perfectly, as they are used constantly:

• (a + b)^2 = a^2 + 2ab + b^2
• (a - b)^2 = a^2 - 2ab + b^2
• a^2 - b^2 = (a + b)(a - b)
• (x + a)(x + b) = x^2 + (a + b)x + ab
• (a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca
• (a + b)^3 = a^3 + b^3 + 3ab(a + b)
• (a - b)^3 = a^3 - b^3 - 3ab(a - b)
• a^3 + b^3 = (a + b)(a^2 - ab + b^2)
• a^3 - b^3 = (a - b)(a^2 + ab + b^2)
• a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)

Special Case for Three Cubes

If a + b + c = 0, then a^3 + b^3 + c^3 = 3abc. This follows directly from the last identity above because when a + b + c = 0, the first factor on the right becomes zero.

Using Identities for Quick Calculation

Identities help us compute values without lengthy multiplication. For example, 103 x 97 = (100 + 3)(100 - 3) = 100^2 - 3^2 = 10000 - 9 = 9991. Similarly, 105^2 = (100 + 5)^2 = 100^2 + 2(100)(5) + 5^2 = 10000 + 1000 + 25 = 11025.

Quick Revision Tables

Table 1: Types of Polynomials

Table 2: Standard Algebraic Identities

Mind Map

Important Diagrams (SVG)

Diagram 1: Graph of a Quadratic Polynomial

Diagram 2: Factorising a Quadratic Using Identities

Common Mistakes

1. Students often write sqrt x or 1/x as polynomials. These are not polynomials because the power of x is not a whole number.
2. A very common error is writing (a + b)^2 = a^2 + b^2, forgetting the middle term 2ab. Always remember the full identity.
3. Students mix up the sign in (a - b)^3. The correct expansion is a^3 - 3a^2b + 3ab^2 - b^3; the last term is negative.
4. When applying the remainder theorem, students forget that dividing by (x + 2) means a = -2, not 2, so they substitute the wrong value.
5. Many students conclude that if p(a) = 0 then (x + a) is a factor. The correct factor is (x - a), because the theorem uses p(a) = 0 implies (x - a) is a factor.
6. Students often mistake the degree of a polynomial like 5 (a constant) as 0 incorrectly; the degree of a non-zero constant polynomial is indeed 0.
7. When factorising a^3 + b^3, students wrongly write (a + b)(a^2 + b^2). The correct second factor is (a^2 - ab + b^2).
8. Students forget the identity a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) and its special case when a + b + c = 0.

Exam Tips

1. Always write the polynomial in the standard form (descending powers of x) before stating its degree, so that the highest power is obvious.
2. For remainder theorem questions, state the theorem and then clearly show p(a); this structured presentation earns full marks.
3. Learn all the identities by heart and practise expanding and factorising with them daily, because they appear in nearly every algebra question.
4. For factorising a cubic polynomial, always first try the factors of the constant term (like +-1, +-2, +-3) to find the first zero quickly.
5. In the identity (a + b + c)^2, carefully write all three square terms and all three double products 2ab, 2bc and 2ca; students usually miss one term.
6. Verify your factorisation by expanding back: if the expansion does not match the original polynomial, you have made a mistake.
7. When using the special case a + b + c = 0 gives a^3 + b^3 + c^3 = 3abc, first explicitly prove that a + b + c = 0 in your solution.

Conclusion

In this chapter we learnt about polynomials: their terms, coefficients and degrees, and the classification into linear, quadratic and cubic polynomials. We studied the concept of the zero of a polynomial and learnt that a polynomial of degree n has at most n zeroes. The remainder theorem gave us a quick way to find remainders when dividing by a linear polynomial, and the factor theorem helped us factorise polynomials completely. Finally, we studied the standard algebraic identities, including the expansions of squares, cubes and the sum and difference of cubes, which simplify computation enormously. These tools will be used in quadratic equations, coordinate geometry and in almost every branch of higher mathematics, so mastering this chapter is essential for every Class 9 student.