Introduction to Linear Polynomials (Class 9): Notes और MCQ Practice

आसान भाषा में

Linear polynomial एक ऐसा univariate (एक variable वाला) algebraic expression होता है जिसमें variable की highest power यानी degree 1 होती है। जब कोई चीज़ हर step पर एक निश्चित मात्रा से बढ़ती है (Linear Growth) या घटती है (Linear Decay), तो उसे linear pattern कहते हैं। इसे equation y = ax + b के रूप में लिखा जाता है, जिसका graph हमेशा एक सीधी रेखा (straight line) बनता है।

Key Points & Formulas

• Degree: Polynomial में variable की उच्चतम power।
• Degree 0: Constant polynomial (जैसे 8 = 8x⁰)
• Degree 1: Linear polynomial (जैसे 3z + 7)
• Degree 2: Quadratic polynomial (जैसे x² + 5x + 1)
• Degree 3: Cubic polynomial (जैसे 5y³ + y² + 2y − 1)
• Linear Polynomial: Degree 1 वाला polynomial। इसके successive integer values का अंतर हमेशा constant रहता है।
• Linear Growth vs Decay: जब quantity समान interval में निश्चित मान से बढ़े तो Linear Growth; यदि निश्चित मान से घटे तो Linear Decay।
• Linear Relationship: y = ax + b।
• Straight Line Graph: y = ax + b का graph हमेशा straight line होता है। Equation y = ax हमेशा origin (0, 0) से गुजरती है (जहाँ a slope है; यदि a > 1 हो तो line y = x से अधिक steep होती है)।

Memory Tricks

1. "Line = 1": Linear polynomial में variable की power हमेशा 1 होती है और इसका graph भी हमेशा एक सीधी Line बनता है।
2. "G for Gain, D for Drop": Linear Growth में value constant amount से Gain (+) होती है, Linear Decay में constant amount Drop (−) होती है।

Common Exam Mistakes

1. Constant की Degree: Constant term (जैसे −9 या 8) की degree को 1 समझ लेना। Constant polynomial की degree हमेशा 0 होती है।
2. Decay में Sign की भूल: Linear decay लिखते समय minus की जगह plus लगा देना (जैसे 100 − 5n को गलती से 100 + 5n लिखना)।

Introduction to Linear Polynomials

1. Chapter in Short

This chapter introduces algebraic expressions in one variable, called univariate polynomials. You will learn how to identify terms, variables, coefficients, and degrees of polynomials, with a special focus on linear polynomials (polynomials of degree 1). The chapter explores how linear polynomials describe real-life situations involving constant addition or subtraction (linear patterns, linear growth, and linear decay), how equating them produces linear equations, how two quantities form a linear relationship of the form y = ax + b, and how to represent these relationships as straight lines on a graph.

2. Key Concepts Explained Simply

Terms, Variables, Coefficients, and Constants

• Variable: A letter used to represent an unknown or changing number (such as x, y, z, m). In earlier grades, these were called letter-numbers.
• Term: The parts of an algebraic expression separated by addition or subtraction signs. For example, in 4x + 5y + 3, the terms are 4x, 5y, and 3.
• Coefficient: The numerical multiplier of a variable. In 4x + 5y + 3, the coefficient of x is 4, and the coefficient of y is 5.
• Constant Term: A term that contains only a fixed number without any variable, such as 3.

Univariate Polynomials and Degree

• An algebraic expression involving only one variable and its non-negative powers is called a one-variable polynomial or univariate polynomial.
• The degree of a polynomial is the highest power of the variable in that polynomial.
• Degree 0 (Constant polynomial): Any non-zero constant, like 8, which can be written as 8x⁰.
• Degree 1 (Linear polynomial): Highest power is 1, like 3z + 7 or 4x.
• Degree 2 (Quadratic polynomial): Highest power is 2, like x² + 5x + 1.
• Degree 3 (Cubic polynomial): Highest power is 3, like 5y³ + y² + 2y − 1.

Polynomial as an Input-Output Process (Function)

• A polynomial can be seen as an input-output machine. If you input a value for the variable x, you get an output value.
• Example: For the polynomial 2x + 3, if the input is x = 4, the output is 2 × 4 + 3 = 11. If the input is x = −6, the output is 2 × (−6) + 3 = −9.

Linear Patterns, Linear Growth, and Linear Decay

• Linear Pattern: A sequence of numbers where the difference between any two consecutive terms is always constant.
• Linear Growth: A situation where a quantity increases by a constant amount over equal intervals.
• Everyday Example: Taxi cost C(d) = 100 + 60d increases by a fixed ₹60 for every 1 km travelled.
• Linear Decay: A situation where a quantity decreases by a constant amount over equal intervals.
• Everyday Example: Water level h(t) = 3 − 0.5t drops by a fixed 0.5 m every month during summer.

Linear Relationships and Graphs

• A linear relationship between two variables x and y can be expressed in the form:

y = ax + b

• When plotted on a coordinate plane, the graph of y = ax + b is always a straight line.
• Any line of the form y = ax passes through the origin (0, 0).
• The number a is called the slope of the line:
• When a > 1, the line is steeper than the line y = x.
• When a < 1 (for a > 0), the line is less steep than the line y = x.
• The line y = x is equally inclined to both axes.

3. Important Definitions, Formulas, and Facts

• Univariate Polynomial: An algebraic expression containing only one variable and its powers.
• Degree of a Polynomial: The highest power of the variable present in the polynomial.
• Classification by Degree:
• Degree 0: Constant polynomial (e.g., 8 = 8x⁰)
• Degree 1: Linear polynomial (e.g., ax + b, where a ≠ 0)
• Degree 2: Quadratic polynomial (e.g., ax² + bx + c)
• Degree 3: Cubic polynomial
• Characteristic Feature of Linear Polynomials: The difference between successive values at integer inputs is constant.
• Linear Equation: An equation formed by equating a linear polynomial to a constant (e.g., 2x + 10 = 64).
• General Linear Relationship: y = ax + b (where a and b are constants).
• Point on a Line: A point (x, y) lies on the line y = ax + b if and only if its coordinates satisfy the equation.

4. Solved Examples

Example 1: Finding Degree, Coefficients, and Constant Term

Question: Consider the polynomial P(x) = x⁴ − 3x³ + 6x² − 2x + 7.

(a) What is the degree of the polynomial?

(b) What are the coefficients of x³ and x²?

(c) What is the constant term?

Answer:

• (a) The highest power of x is 4, so the degree is 4.
• (b) The coefficient of x³ is −3, and the coefficient of x² is 6.
• (c) The constant term is 7.

Example 2: Word Problem on Linear Equations

Question: Ruby has 3 times as many two-rupee coins as she has five-rupee coins. If she has a total of ₹88, how many coins does she have of each type?

Answer:

• Let the number of 5-rupee coins be x.
• Then, the number of 2-rupee coins = 3x.
• Value of 5-rupee coins = 5 × x = 5x rupees.
• Value of 2-rupee coins = 2 × (3x) = 6x rupees.
• Total value = 5x + 6x = 11x rupees.
• We are given that the total amount is ₹88:

11x = 88

x = 88 ÷ 11 = 8

• Therefore, Ruby has:
• 5-rupee coins = 8
• 2-rupee coins = 3 × 8 = 24

Example 3: Finding a Linear Relationship (y = ax + b)

Question: A learning platform charges a fixed monthly fee and an additional cost per digital module accessed. A student observes that when 10 modules are accessed, the bill is ₹400. When 14 modules are accessed, the bill is ₹500. If the monthly bill y depends on the number of modules x by the relation y = ax + b, find the values of a and b.

Answer:

• When x = 10, y = 400:

400 = 10a + b ...(Equation 1)

• When x = 14, y = 500:

500 = 14a + b ...(Equation 2)

• From Equation 1, express b in terms of a:

b = 400 − 10a

• Substitute this into Equation 2:

500 = 14a + (400 − 10a)

500 = 4a + 400

4a = 500 − 400 = 100

a = 100 ÷ 4 = 25

• Now find b:

b = 400 − 10 × 25 = 400 − 250 = 150

• Thus, a = 25 and b = 150 (and the equation is y = 25x + 150).

Example 4: Linear Growth and Linear Decay

Question:

(a) A prepaid mobile balance starts at ₹600 and reduces by ₹15 each day. Write an equation for the balance b(x) after x days and find after how many days the balance will run out.

(b) Does this represent linear growth or linear decay?

Answer:

• (a) The equation modeling the remaining balance after x days is:

b(x) = 600 − 15x

The balance runs out when b(x) = 0:

600 − 15x = 0

15x = 600

x = 600 ÷ 15 = 40 days.

• (b) It represents linear decay because the balance decreases by a constant amount (₹15) over equal intervals of time (each day).

5. Common Mistakes and Quick Revision Points

• Degree of a Constant: Remember that a non-zero constant like −9 or 8 has a degree of 0 because it can be written as 8x⁰. It is NOT degree 1.
• Missing Signs in Coefficients: Always include the negative sign with a coefficient. For example, in 4z³ + 5z² − 11, the coefficient of z is 0 (since there is no z term), and the constant term is −11, not 11.
• Linear Growth vs. Linear Decay:
• Growth = quantity increases by a constant amount each step (+a).
• Decay = quantity decreases by a constant amount each step (−a).
• Plotting a Line: You need at least two distinct points to draw a straight line. Always check your points by substituting them back into y = ax + b.
• Origin Rule: Any line given by y = ax will always pass through the origin (0, 0).