Introduction to Linear Polynomials · Lesson 1 of 7
Introduction
“Meet the straight-line family: simple rules with surprisingly eventful graphs.”
• Recall the meaning of algebraic expressions, terms, variables, coefficients and constants. • Distinguish expressions involving one variable from expressions involving more than one variable. • Understand what a polynomial is and how its degree is determined. • Recognise constant, linear, quadratic and cubic polynomials. • Connect algebraic expressions with real situations.
Introduction
Algebra gives us a clear and compact way to describe quantities that can change. Instead of writing long statements again and again, we use letters, numbers, and mathematical operations to form algebraic expressions. These expressions help us represent relationships in a simple form and make it easier to study patterns and solve problems.
Before understanding linear polynomials, it is important to be comfortable with the basic language of algebra. Ideas such as terms, variables, coefficients, constants, and degree tell us how an expression is built and how it should be interpreted. In this chapter, we will use these ideas to understand what makes a polynomial linear and how linear polynomials behave.
From a Situation to an Algebraic Expression
Suppose red boxes contain 4 pens each and blue boxes contain 5 pencils each. If x red boxes and y blue boxes are bought, then the numbers of pens and pencils are 4x and 5y. If 3 extra pens are also received, the total number of items is represented by 4x + 5y + 3.
| Part of 4x + 5y + 3 | Meaning |
|---|---|
| 4x, 5y, 3 | Terms |
| x, y | Variables |
| 4, 5 | Coefficients |
| 3 | Constant term |
A variable is a symbol, usually a letter, that can take different numerical values.
A coefficient is the numerical factor multiplying a variable or a power of a variable.
A constant is a term with no variable attached to it.
Expressions Can Involve More Than One Variable
Consider a rectangular garden with length l and width w. If wire fencing along the two lengths costs ₹100 per metre, fencing along the two widths costs ₹80 per metre, and seeds cost ₹50 per square metre, then the total cost becomes 200l + 160w + 50lw. This expression has two variables and also contains the product lw.
Problem
Explain how the expression 200l + 160w + 50lw is formed.
- 1.Two lengths together measure 2l, so wire fencing costs 2l × 100 = 200l.
- 2.Two widths together measure 2w, so wooden fencing costs 2w × 80 = 160w.
- 3.The garden area is lw, so seeds cost 50lw.
- 4.Adding all parts gives 200l + 160w + 50lw.
A One-Variable Expression from Geometry
Now take a wire of length 20 cm and bend it into a rectangle. If one side is x cm, the other side must be 10 − x cm because the perimeter is 20 cm. The area is therefore x(10 − x) = 10x − x². This expression contains only one variable, x, but it contains more than one power of x.
One-Variable Polynomials
Expressions such as 4x, x² + 1, 2y − 5, 5y³ + y² + 2y − 1 and 3z + 7 involve only one variable. Such expressions are called one-variable polynomials or univariate polynomials.
A polynomial involving only one variable and non-negative whole-number powers of that variable is called a univariate polynomial.
Degree of a Polynomial
The degree tells us the highest power of the variable that actually appears with a non-zero coefficient. It is one of the most important ways to classify polynomials.
The degree of a polynomial is the highest power of its variable with a non-zero coefficient.
| Polynomial | Highest power | Degree | Name |
|---|---|---|---|
| 8 | 0 | 0 | Constant polynomial |
| 3z + 7 | 1 | 1 | Linear polynomial |
| x² + 5x + 1 | 2 | 2 | Quadratic polynomial |
| 5y³ + y² + 2y − 1 | 3 | 3 | Cubic polynomial |
A non-zero constant such as 8 can be written as 8x⁰. Since x⁰ = 1, the expression is still 8, and its degree is 0.
Problem
For p(x) = x⁴ − 3x³ + 6x² − 2x + 7, identify the coefficients of x⁴, x³, x² and x, and the constant term.
- 1.Coefficient of x⁴ = 1.
- 2.Coefficient of x³ = −3.
- 3.Coefficient of x² = 6.
- 4.Coefficient of x = −2.
- 5.Constant term = 7.
- 6.The highest power is 4, so the degree is 4.
Practice Problems
- Find the degree of 2x² − 5x + 3.
- Find the degree of y³ + 2y − 1.
- Find the degree of −9.
- Find the degree of 4z − 3.
- Write one polynomial each of degrees 1, 2 and 3.
- In x⁴ − 3x³ + 6x² − 2x + 7, identify the coefficients of x² and x³.
- Identify the coefficient of z in 4z³ + 5z² − 11.
- Identify the constant term in 9x³ + 5x² − 8x − 10.
Key Takeaways
• Algebraic expressions are built from terms, variables, coefficients, constants and operations. • A univariate polynomial contains only one variable. • The degree is the highest power of the variable with non-zero coefficient. • Degree 1 polynomials are linear, degree 2 are quadratic and degree 3 are cubic. • Real situations can often be translated naturally into algebraic expressions.
Next, we focus specifically on degree-1 polynomials and learn why their values change at a constant rate.
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Linear Polynomials