What is a Polynomial of Degree One?

S3-SA1-0343

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

A polynomial of degree one is an algebraic expression where the highest power of the variable is 1. It is also known as a linear polynomial because when you graph it, you get a straight line.

Simple Example

Quick Example

Imagine you are buying samosas. Each samosa costs Rs 10. If 'x' is the number of samosas you buy, the total cost would be 10 times 'x', which is 10x. Here, the highest power of 'x' is 1, so 10x is a polynomial of degree one.

Worked Example

Step-by-Step

Question: Is 3x + 5 a polynomial of degree one? Let's check.
---Step 1: Identify the variable in the expression. The variable here is 'x'.
---Step 2: Look at the power (exponent) of the variable 'x'. In '3x', 'x' has a power of 1 (it's like x^1, but we usually don't write the 1).
---Step 3: Check if there are any other terms with variables. Here, '5' is a constant term and does not have a variable 'x' with a power higher than 1.
---Step 4: Since the highest power of the variable 'x' is 1, the expression 3x + 5 is indeed a polynomial of degree one.
Answer: Yes, 3x + 5 is a polynomial of degree one.

Why It Matters

Understanding polynomials of degree one helps us model simple relationships in the real world, like how distance changes with time or how much a bill increases with more items. Engineers use them to design simple circuits, and economists use them to understand basic market trends, helping them make predictions.

Common Mistakes

MISTAKE: Thinking that 2x^2 + 3 is a polynomial of degree one because it has 'x' in it. | CORRECTION: The degree is determined by the *highest* power of the variable. In 2x^2 + 3, the highest power is 2 (from x^2), so it's a polynomial of degree two, not one.

MISTAKE: Confusing constants with variables when checking degree, e.g., thinking 5y has a degree of 5. | CORRECTION: The degree is based on the power of the *variable* (like x, y, z), not the number multiplied with it (the coefficient). The degree of 5y is 1 because 'y' has a power of 1.

MISTAKE: Assuming an expression like 7 is a polynomial of degree one. | CORRECTION: 7 is a constant. It can be written as 7x^0, so its degree is 0. A polynomial of degree one *must* have a variable raised to the power of 1.

Practice Questions

Try It Yourself

QUESTION: Is 4y - 2 a polynomial of degree one? | ANSWER: Yes

QUESTION: Which of these is NOT a polynomial of degree one: a) 9z, b) 5 + m, c) p^2 - 1, d) 100 - 3k | ANSWER: c) p^2 - 1

QUESTION: A taxi charges Rs 20 for every kilometer traveled. If 'd' is the distance in kilometers, write an expression for the total charge. Is this a polynomial of degree one? | ANSWER: The expression is 20d. Yes, it is a polynomial of degree one.

MCQ

Quick Quiz

Which of the following expressions is a polynomial of degree one?

x^3 + 2

5y - 7

4x^2

The Correct Answer Is:

Option B (5y - 7) has 'y' as the variable with the highest power of 1. Option A has x^3 (degree 3), Option C has x^2 (degree 2), and Option D is a constant (degree 0).

Real World Connection

In the Real World

Think about your mobile phone data plan. If you pay a fixed amount plus an extra charge for every GB of data you use beyond a limit, that relationship can be described by a polynomial of degree one. Companies like Jio or Airtel use these linear models to calculate your monthly bill, showing how cost changes directly with data usage.

Key Vocabulary

Key Terms

POLYNOMIAL: An expression with variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents of variables. | DEGREE: The highest power of the variable in a polynomial. | VARIABLE: A symbol (like x, y) that represents a quantity that can change. | COEFFICIENT: A number multiplied by a variable in an algebraic expression. | CONSTANT: A number whose value does not change.

What's Next

What to Learn Next

Great job understanding polynomials of degree one! Next, you can explore 'What is a Polynomial of Degree Two?' This will help you understand more complex relationships and curves, which are super important in subjects like physics and engineering. Keep up the good work!