Introduction to Polynomials

What is a Polynomial?

The chapter Polynomials is one of the most important topics in Class 10 Mathematics, and it forms the foundation for algebra in higher classes. A polynomial in x is an algebraic expression made up of a finite number of terms, where each term has the form "a multiplied by x raised to the power n", with the condition that the coefficient a is a real number and the exponent n is a whole number (0, 1, 2, 3, and so on).

For example, expressions like 3x² − 1, 7x − 1/2, and x³ − x² + x − 1 are all polynomials because every power of x is a whole number. However, expressions such as 3/x² − 1, x²⁰ − 5x⁻³, and 1/x − 1 are not polynomials, because they contain negative powers or x in the denominator, which violates the whole-number exponent rule. Recognizing this difference is one of the first skills students must master in this chapter.

Degree of a Polynomial

The degree of a polynomial p(x) is defined as the highest power of x that appears in the expression. The degree determines the name and behavior of the polynomial, and it is a concept used repeatedly throughout the chapter, especially when studying the shapes of graphs and the number of zeroes a polynomial can have.

• Zero polynomial — the constant 0, whose degree is not defined
• Constant polynomial — a non-zero constant such as −5, with degree 0
• Linear polynomial — degree 1, general form ax + b, such as 3x − 5
• Quadratic polynomial — degree 2, general form ax² + bx + c, such as 2y² − y − 3
• Cubic polynomial — degree 3, general form ax³ + bx² + cx + d
• Biquadratic polynomial — degree 4, such as 7z⁴ − 3z²

Finding the Value of a Polynomial

The value of a polynomial p(x) at x = k is found by substituting k in place of x throughout the expression and simplifying. This value is written as p(k). This concept is essential because it is used to verify zeroes, solve equations, and check answers in later parts of the chapter.

For example, if p(x) = x² − 2, then substituting x = 2 gives p(2) = 2² − 2 = 4 − 2 = 2. Similarly, if f(y) = y³ + 8y + 1, then f(3) = 27 + 24 + 1 = 52. Practicing such substitutions with both positive and negative values of x builds accuracy, since sign errors are one of the most common mistakes students make in this topic.

Worked Example: Evaluating a Quadratic Polynomial

Consider the polynomial p(x) = x² − 5x − 6. To find its value at several points, substitute each value of x and simplify step by step. For instance, p(1) = (1)² − 5(1) − 6 = 1 − 5 − 6 = −10, and p(−1) = (−1)² − 5(−1) − 6 = 1 + 5 − 6 = 0. Notice that when x = −1, the result is exactly 0 — this special case leads directly to the next important concept in the chapter.

By calculating p(0), p(1), p(2), p(3), and the corresponding negative values, students can observe how the output changes as x changes, which builds intuition for graphing polynomials later in the chapter.

Zeroes of a Polynomial

A real number k is called a zero of the polynomial p(x) if p(k) = 0. In other words, a zero is a value of x that makes the entire polynomial equal to zero when substituted. Finding zeroes is one of the central goals of this chapter, since zeroes correspond to the points where the graph of the polynomial crosses the x-axis.

For example, if p(x) = x² − 4, then p(2) = 2² − 4 = 0 and p(−2) = (−2)² − 4 = 0, so both 2 and −2 are zeroes of this polynomial. Similarly, for p(x) = x² − 4x + 3, substituting x = 1 and x = 3 both give p(x) = 0, so 1 and 3 are the zeroes of this quadratic polynomial. A quadratic polynomial can have at most two zeroes.

Common Mistakes to Avoid

• Confusing expressions with negative or fractional exponents with valid polynomials — only whole-number powers of the variable are allowed
• Forgetting to apply the correct sign when substituting negative values of x into a polynomial
• Mixing up the "degree" of a polynomial with the "number of terms" — degree refers only to the highest power
• Assuming every value of x gives a zero, instead of checking whether p(k) actually equals 0
• Writing the general form of a polynomial without ensuring the leading coefficient a₀ is not zero

What This Lesson Prepares You For

This introduction lays the groundwork for the rest of the Polynomials chapter, where students learn to find the relationship between zeroes and coefficients, divide polynomials, and apply the factor theorem. A strong understanding of degree, value, and zeroes is also essential for solving quadratic equations later in the syllabus.

Students following the CBSE, Telangana, and Andhra Pradesh board syllabus should also revisit set notation from the previous chapter, especially the idea of equal sets and subsets, since similar logical reasoning is used while comparing solutions and zeroes of polynomials.