Polynomials

2.1 INTRODUCTION

In this section, the concept of algebraic expressions is revisited with a focus on a special type called linear polynomials. Algebraic expressions combine variables (also called letter-numbers) and coefficients using operations such as addition, subtraction, and multiplication. The section begins with practical examples to illustrate how algebraic expressions model real-life situations. For instance, if Raju buys x red boxes each containing 4 pens and y blue boxes each containing 5 pencils, plus 3 extra pens, the total number of pens and pencils can be expressed as 4x + 5y + 3. Here, 4x, 5y, and 3 are terms; x and y are variables; 4 and 5 are coefficients; and 3 is a constant term. Another example involves calculating the total cost of fencing and decorating a rectangular garden with length l and width w. The cost expression is 200l + 160w + 50lw, where each term corresponds to a specific cost component. A third example considers a wire bent to form rectangles with length x and width (10 - x), leading to an expression for area as x(10 - x) or 10x - x². This illustrates how algebraic expressions can involve powers of variables. The section concludes by defining univariate polynomials—algebraic expressions in one variable with non-negative integer powers—and introduces the concept of degree, which is the highest power of the variable. Examples of constant (degree 0), linear (degree 1), quadratic (degree 2), and cubic (degree 3) polynomials are given to familiarize students with terminology.

• Algebraic expressions combine variables and coefficients using addition, subtraction, and multiplication.
• Variables are letters representing numbers; coefficients are numerical factors of variables.
• Terms are parts of an expression separated by plus or minus signs.
• Univariate polynomials involve only one variable with non-negative integer powers.
• Degree of a polynomial is the highest power of the variable in the expression.
• Examples of polynomials: constant (degree 0), linear (degree 1), quadratic (degree 2), cubic (degree 3).
• 📌 Variable: A symbol representing a number, e.g., x, y.
• 📌 Coefficient: Numerical factor multiplying a variable, e.g., 4 in 4x.
• 📌 Term: A part of an algebraic expression separated by + or - signs.

2.2 LINEAR POLYNOMIALS

This section introduces linear polynomials, which are polynomials of degree one. It starts with examples such as the perimeter of a square with side x, which is 4x, a linear polynomial. Another example models a chess club fee structure where a joining fee of ₹200 plus ₹50 per match played leads to the total cost expression 200 + 50m, a linear polynomial in variable m (number of matches). The section highlights the characteristic feature of linear polynomials: the difference between successive values at integer points is constant, representing linear patterns. It also explains how linear polynomials relate to linear equations when equated to constants, demonstrated by solving a problem involving two numbers with a given sum and difference. The concept of polynomials as input-output functions is introduced, exemplified by the linear polynomial 2x + 3, where substituting values of x yields corresponding outputs. The section contrasts linear functions with quadratic functions, such as 10x - x². The input-output process is visualized as a function machine, emphasizing the functional nature of polynomials.

• Linear polynomials have degree 1 and are of the form ax + b.
• Examples include perimeter of square (4x) and cost expressions like 200 + 50m.
• Linear polynomials show constant difference between successive integer values.
• Equating linear polynomials to constants leads to linear equations.
• Polynomials can be viewed as functions mapping inputs to outputs.
• Linear functions produce outputs increasing or decreasing at a constant rate.
• 📌 Linear polynomial: Polynomial of degree one, e.g., 2x + 3.
• 📌 Linear equation: Equation formed by equating a linear polynomial to a constant.
• 📌 Function: A relation where each input has a unique output.