Linear Polynomials

Class 09 Maths

An algebraic expression combines numbers, variables, and ­operation symbols. For example, 2x2 + 5xy - 3y2 is an algebraic expression in the variables x and y. 2x2, 5xy and -3y2 are the terms of the algebraic expression, and the numbers 2, 5 and -3 are coefficients of the terms.

Expressions such as 4x, x2 + 1, 2y - 5, 5y3 + y2 + 2y - 1, 3z + 7 are algebraic expressions that involve only one variable: x, y or z.

In an algebraic expression, the powers of a variable also appear. For example, in the expression x2 + 5x + 1, the highest power of x is 2.

Algebraic expressions involving one variable and its powers are called one-variable polynomials or univariate polynomials. The highest power of the variable in a polynomial is called its degree.

  1. 5y3 + y2 + 2y - 1 is a polynomial of degree 3. Such polynomials are called cubic polynomials.
  2. x2 + 5x + 1 is a polynomial of degree 2. Such polynomials are called quadratic polynomials.
  3. 3z + 7 is a polynomial of degree 1. Such polynomials are called linear polynomials.
  4. The constant 8 is a polynomial of degree 0 as it can be written as 8x0 in which the power of the variable x is 0. Such polynomials are called constant polynomials.

Linear Equation

When we equate a linear polynomial in one variable to a constant, we get a linear equation.

Example: The sum of two numbers is 64. One of the numbers is 10 more than the other. What are the two numbers?

Let the smaller number be x. Then the larger number must be x + 10.

Since their sum is 64, x + (x + 10) = 64.

This implies that 2x + 10 = 64.

2x = 54 or x = 27.

The numbers are 27 and 37.

Linear Pattern

A linear pattern is a sequence of numbers where the difference between two consecutive terms is constant.

Linear growth refers to a pattern in which a quantity increases by a fixed amount over equal intervals. In contrast, linear decay describes a pattern in which a quantity decreases by a fixed amount over equal intervals.

A linear relationship between two variables x and y is ­represented by a straight line y = ax + b. The slope of this line is a. The ­constant b is called the y-intercept which is the distance from the origin where the line cuts the y-axis. When b = 0, the equation of the line becomes y = ax and the line passes through the origin.

Linear growth is represented by a straight line with positive slope and linear decay is represented by a straight line with ­negative slope.

Parallel lines are of the form y = ax + b, where the slope a is fixed while b, the y-intercept, varies.