An **algebraic expression** is a combination of variables, constants, and arithmetic operations. Variables are letter-symbols (like x, y, z) that represent unknown or changing quantities. Constants are fixed numerical values.
**Components of algebraic expressions:**
**Real-life example:** If Raju buys x red boxes with 4 pens each and y blue boxes with 5 pencils each, plus 3 free pens, the total is 4x + 5y + 3. Here, x and y are variables, 4 and 5 are coefficients, 3 is a constant.
**Exam point:** Always identify terms, coefficients, and constants when analyzing expressions.
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A **polynomial** is an algebraic expression involving one variable and powers of that variable with real number coefficients. The general form is: a₀ + a₁x + a₂x² + ... + aₙxⁿ, where aᵢ are coefficients and n ≥ 0.
**Key characteristic:** In a polynomial, the powers of variables must be whole numbers (non-negative integers). Expressions like 1/x or √x are NOT polynomials.
**Degree of a polynomial:** The **degree** is the highest power of the variable in the polynomial.
**Classification by degree:**
**Identification method:** Look at the highest power of the variable present in the polynomial.
**Example problems:**
**Coefficient identification:** In x⁴ – 3x³ + 6x² – 2x + 7, the coefficient of x² is 6, coefficient of x³ is –3, and constant term is 7.
**Exam note:** Questions often ask to identify degree and coefficients. Write polynomials clearly with all terms.
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A **linear polynomial** is a polynomial of degree 1, expressed in the form **ax + b** where a ≠ 0, a and b are real numbers, and x is a variable.
**Characteristics of linear polynomials:**
**Key property – Constant difference:** When a linear polynomial is evaluated at consecutive integer values of the variable, the differences are always the same.
**Example:** For polynomial 4x, when x = 1, 2, 3, 4:
**Practical example:** Perimeter of square with side x is 4x (linear polynomial). For squares with sides 1, 1.5, 2, 2.5, 3 cm:
**Chess club example:** Club charges ₹200 joining fee + ₹50 per match. Total cost for m matches: **200 + 50m**
Difference is constant ₹50 per match.
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When a linear polynomial is equated to a constant, we get a **linear equation**: **ax + b = c**
**Method to solve:**
**Example:** Two numbers sum to 64; one is 10 more than other. Find both.
**Verification:** 27 + 37 = 64 ✓ and 37 = 27 + 10 ✓
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A polynomial represents an **input-output process** where each input value of the variable produces exactly one output value.
**Process:**
**Example:** For linear polynomial 2x + 3:
**Rectangle example:** Wire bent to form rectangles with perimeter 20 cm. If length = x cm, then width = (10 – x) cm. Area = x(10 – x) = 10x – x². This is a quadratic function (not linear).
**Linear vs. Quadratic function:** 2x + 3 is linear (degree 1), but 10x – x² is quadratic (degree 2).
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A **linear pattern** is a sequence where the difference between consecutive terms is constant.
**Growing tile pattern example:**
**General formula:** For stage n, number of tiles = **2n – 1**
**Pocket money decay example:** Bela has ₹100, spends ₹5 daily. Amount left after n days = **100 – 5n**
To find when ₹40 remains: 100 – 5n = 40 → 5n = 60 → n = 12 days
**Auto-rickshaw fare example:** Base fare ₹25 for 2 km, then ₹15/km. For n km (where n ≥ 2): Fare = **25 + 15(n – 2) = 15n – 5**
**Exam technique:** Identify the initial value and constant change per unit. Write formula as f(n) = initial + (constant change)(n).
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**Linear growth** describes a linear pattern where a quantity increases by a constant amount over equal intervals.
**Example:** Journey cost C(d) = 100 + 60d rupees, where d is distance in km
**Linear decay** describes a pattern where a quantity decreases by a constant amount over equal intervals.
**Example:** Water height in tank h(t) = 3 – 0.5t meters, where t is months
**Key distinction:**
**Real-life applications:** Plant growth (growth), phone value depreciation (decay), population migration (growth), balance reduction (decay), water tank emptying (decay).
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A **linear relationship** between two variables x and y is expressed as **y = ax + b**, where:
**Method to find linear relationship:**
1. Identify two data points (x₁, y₁) and (x₂, y₂)
2. Create two equations by substituting into y = ax + b
3. Solve the system of equations for a and b
**Example:** Telecom bill depends on GB of data used. Bill is ₹350 for 10 GB and ₹550 for 20 GB. Find relationship y = ax + b.
Given points: (10, 350) and (20, 550)
Equations:
Subtract (1) from (2):
Substitute a = 20 in equation (1):
**Linear relationship: y = 20x + 150**
**Interpretation:** ₹20 per GB (slope) + ₹150 fixed monthly fee (intercept)
**Temperature conversion example:** 0°C = 32°F (ice melts) and 100°C = 212°F (water boils). Find °C = a(°F) + b.
Points: (32, 0) and (212, 100)
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Linear relationships are visualized by plotting the equation y = ax + b as a straight line on a coordinate plane.
**Steps to plot:**
1. Find two points on the line by choosing x-values and calculating corresponding y-values
2. Plot the two points on coordinate axes
3. Draw a straight line through both points
**Finding points on y = 2x + 1:**
**Graphical properties:**
**Real-world interpretation:**
**Exam preparation:** Questions may ask to plot equations, find intercepts, or interpret graphs. Always label axes clearly and mark at least two points.
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**Basic polynomial form:** a₀ + a₁x + a₂x² + ... + aₙxⁿ
**Linear polynomial:** ax + b (degree = 1)
**Linear equation:** ax + b = c
**Linear function:** f(x) = ax + b or y = ax + b
**Evaluating polynomial:** Substitute variable value into expression
**Linear growth:** y = initial + rate × x (a > 0)
**Linear decay:** y = initial – rate × x (a < 0)
**Constant difference property:** For linear patterns, consecutive term difference = slope value
**Exam Success Tips:**
Q1. What is the degree of the polynomial x⁴ - 3x³ + 6x² - 2x + 7?
Answer: A — The degree is determined by the highest power of the variable, which is x⁴, so degree = 4.
Q2. Which of the following is a linear polynomial?
Answer: B — 3z + 7 has degree 1, making it a linear polynomial; the others have degrees 2, 3, and 2 respectively.
Q3. What is the constant term in the polynomial 9x³ + 5x² - 8x - 10?
Answer: C — The constant term is the term without any variable, which is -10.
Q4. A chess club charges ₹200 joining fee plus ₹50 per match. If a member played m matches, the total cost is represented by the polynomial 200 + 50m. What will be the cost for 5 matches?
Answer: D — Substituting m = 5: 200 + 50(5) = 200 + 250 = ₹550.
Q5. The perimeter of a square with side x cm is 4x cm. What is the perimeter when x = 2.5 cm?
Answer: B — Perimeter = 4x = 4(2.5) = 10 cm.
Q6. In the polynomial 3x² + 5x - 2, what is the coefficient of x?
Answer: B — The coefficient of x is the number directly multiplying x, which is 5.
Q7. Which of the following is NOT a polynomial?
Answer: C — 5x⁻¹ + 2 has a negative power of x, so it is not a polynomial (polynomials require non-negative integer powers only).
Q8. Ramesh observes that in an arithmetic sequence 5, 8, 11, 14, 17, ..., the difference between consecutive terms is always 3. Which polynomial property does this demonstrate?
Answer: B — A constant difference of 3 between successive values demonstrates the linear pattern property of polynomials.
Q9. If 2x + 10 = 64, then the value of x is:
Answer: A — Solving: 2x + 10 = 64 → 2x = 54 → x = 27.
Q10. A rectangular garden has length l metres and width w metres. If the expression for total cost of fencing is 200l + 160w + 50lw, which term makes this NOT a polynomial in one variable?
Answer: C — The term 50lw contains two variables (l and w), so the entire expression is a polynomial in two variables, not one variable.
What is a polynomial in one variable?
An algebraic expression involving one variable and its non-negative integer powers with real coefficients.
Define the degree of a polynomial.
The highest power of the variable that appears in the polynomial with a non-zero coefficient.
What is a linear polynomial?
A polynomial of degree 1, written in the form ax + b where a ≠ 0 and a, b are real numbers.
What is the coefficient of x in the polynomial 5x² + 3x - 7?
The coefficient of x is 3.
Identify the constant term in 4z³ + 5z² - 11.
The constant term is -11.
What is a linear pattern?
A sequence where the difference between successive values at integers is constant.
What do we get when we equate a linear polynomial to a constant?
A linear equation that can be solved to find the value of the variable.
Write a linear polynomial with variable x and constant term 7.
Any polynomial of the form ax + 7 where a ≠ 0, such as 2x + 7 or 3x + 7.
What is the degree of the constant polynomial 8?
The degree is 0 because it can be written as 8x⁰.
How can we interpret a polynomial as an input-output machine?
Input the value of the variable, substitute it in the polynomial expression, and the result is the output.
Define a linear polynomial and give two examples. [2 marks]
Linear polynomial has degree 1 with form ax + b (a ≠ 0). Examples: 3x + 5, 2z - 7, etc.
The sum of two consecutive even numbers is 50. Write the linear equation and find the two numbers. [3 marks]
Let the first even number be x, then the second is x + 2. Set up equation: x + (x + 2) = 50, then solve for x.
A wire of length 30 cm is bent to form a rectangle. If one side is x cm, write the expression for the area of the rectangle and determine the value of x for which the area is maximum. Explain your reasoning. [5 marks]
Perimeter = 30, so 2(x + width) = 30, giving width = 15 - x. Area = x(15 - x) = 15x - x². This is a quadratic (not linear), but you can test values or recognize that the polynomial 15x - x² achieves maximum when x = 7.5 cm by symmetry or calculus reasoning.
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