Study Notes on Linear Polynomials

Algebraic expressions are combinations of numbers, variables, and operation symbols.
Example 1: Raju’s Shopping Scenario
- Red boxes contain 4 pens each ( $x$ boxes).
- Blue boxes contain 5 pencils each ( $y$ boxes).
- Raju receives 3 extra pens for free.
- The algebraic expression representing the total quantity is $4x + 5y + 3$ .
Components of the Expression $4x + 5y + 3$ :
- Terms: $4x$ , $5y$ , and $3$ .
- Variables (formerly called letter-numbers): $x$ and $y$ .
- Coefficients: The numbers $4$ and $5$ are the coefficients of $x$ and $y$ respectively.
- Constant: The number $3$ is a fixed value that does not involve a variable.

Definition of Univariate (One-Variable) Polynomials: These are algebraic expressions involving only one variable and its powers (e.g., $x, y, \text{ or } z$ ).
The Degree of a Polynomial: This is defined as the highest power of the variable present in the polynomial.
Classification of Polynomials by Degree:
- Constant Polynomial: Degree 0. Example: The number $8$ can be written as $8x^0$ .
- Linear Polynomial: Degree 1. Example: $3z + 7$ .
- Quadratic Polynomial: Degree 2. Example: $x^2 + 5x + 1$ .
- Cubic Polynomial: Degree 3. Example: $5y^3 + y^2 + 2y - 1$ .
Discussion Limitation: The focus of this chapter is strictly on algebraic expressions involving one variable.

Linear polynomials are polynomials of degree 1.
Example 4: Perimeter of a Square
- The perimeter of a square with side length $x$ is given by $4x$ , which is a linear polynomial.
Example 5: Chess Club Fees
- Joining fee: $`200$ .
- Fee per match played: $`50$ .
- For $m$ matches, the total cost is $200 + 50m$ .
- This represents a linear pattern because the cost increases by a constant value ( $`50$ ) for every additional match.
Polynomials as Input-Output Processes (Functions):
- A linear polynomial like $2x + 3$ can be viewed as an input-output machine.
- If input $x = 4$ , output is $2 \times 4 + 3 = 11$ .
- If input $x = -6$ , output is $2 \times (-6) + 3 = -9$ .

Linear Equations: Formed when a linear polynomial in one variable is equated to a constant (e.g., $2x + 10 = 64$ ).
Linear Patterns: Sequences where the difference between two consecutive terms is constant.
Linear Growth: Occurs when a quantity increases by a fixed amount over equal intervals.
- Example 9: Journey Cost: $C(d) = 100 + 60d$ . For every 1 km increase in distance $d$ , cost $C$ increases by $`60$ .
Linear Decay: Occurs when a quantity decreases by a fixed amount over equal intervals.
- Example 10: Summer Water Tank Level: Initial height is $3\,m$ . Monthly decrease is $0.5\,m$ . The linear function is $h(t) = 3 - 0.5t$ .

A linear relationship between variables $x$ and $y$ is expressed as $y = ax + b$ .
Example 11: Telecom Billing (Finding $a$ and $b$ ):
- Bill is $`350$ for $10\,GB$ .
- Bill is $`550$ for $20\,GB$ .
- System of equations:
  - $350 = 10a + b$
  - $550 = 20a + b$
- Solving for $a$ and $b$ :
  - From the first equation: $b = 350 - 10a$ .
  - Substitute into second: $550 = 20a + (350 - 10a) \rightarrow 550 = 10a + 350 \rightarrow 10a = 200 \rightarrow a = 20$ .
  - Calculate $b$ : $b = 350 - 200 = 150$ .
- Final relationship: $y = 20x + 150$ .

To plot a line $y = ax + b$ , identify at least two points $(x, y)$ .
The Origin (0,0): Lines of the form $y = ax$ (where $b = 0$ ) always pass through the origin.
Slope ( $a$ ): Represents the steepness of the line.
- If a > 1, the line is steeper than $y = x$ .
- If a < 1, the line is less steep than $y = x$ .
- Positive slope (a > 0) represents linear growth.
- Negative slope (a < 0) represents linear decay.
Y-intercept ( $b$ ): The coordinate point $(0, b)$ where the line cuts the y-axis.
- If $b = 3$ , the line cuts the y-axis 3 units above the origin.
- If $b = -2$ , the line cuts the y-axis 2 units below the origin.
Parallel Lines: Lines that have equal slopes ( $a$ ) but different y-intercepts ( $b$ ) are parallel to each other. Changing $b$ while keeping $a$ fixed shifts the line vertically.

Sequence Pattern Example: Square tile pattern stages 1, 3, 5, 7…
- Rule: Stage $n$ has $2n - 1$ tiles.
- This is a linear polynomial of degree 1 with a constant difference of $2$ .
Fare Calculation Example: Auto fare is $`25$ for first 2 km, then $`15$ per km.
- For $n \ge 2$ , Fare $= 25 + 15 \times (n - 2) = 15n - 5$ .
Temperature Conversion: The relationship between Celsius ( $^\circ C$ ) and Fahrenheit ( $^\circ F$ ) is given by $^\circ C = a ^\circ F + b$ .
- Water melts at $0 ^\circ C$ (32 $^\circ F$ ) and boils at $100 ^\circ C$ (212 $^\circ F$ ).
Kelvin to Fahrenheit Conversion: $y = \frac{9}{5}(x - 273) + 32$ .
Physics Application (Work Done): Work ( $w$ ) is the product of constant force and distance ( $d$ ). If force is $3$ units, then $w = 3d$ .

Think and Reflect (Page 2):
- Identify terms, variables, and coefficients in total cost $200l + 160w + 50lw$ :
  - Terms: $200l, 160w, 50lw$ .
  - Variables: $l$ and $w$ .
  - Coefficients: $200, 160, 50$ .
- Difference from Example 1? Example 1 resulted in a linear sum ( $4x + 5y + 3$ ), while Example 2 includes a product term ( $50lw$ ).
Think and Reflect (Page 2 - Area):
- Wire of $20\,cm$ length bent into rectangle: if length is $x$ , width is $10 - x$ . Area is $10x - x^2$ .
- Similarity or difference between Example 1 and 3? Example 1 used two variables, whereas Example 3 uses only one variable but has a squared term ( $x^2$ ).
Think and Reflect (Page 4):
- If a chess player paid $`750$ , how many matches did they play?
  - $200 + 50m = 750 \rightarrow 50m = 550 \rightarrow m = 11$ matches.
Think and Reflect (Page 12):
- What do $20$ and $150$ represent in $y = 20x + 150$ ?
  - $20$ represents the additional cost per GB of data used.
  - $150$ represents the fixed monthly fee.
Think and Reflect (Page 21):
- What do functions of the form f(x) = ax + a, a > 0 have in common? They all have the same value for slope and y-intercept ( $a$ ), and all pass through the point $(-1, 0)$ .