01 · Explore
Understanding Algebraic Expressions
Algebraic expressions are the building blocks of algebra, combining numbers and variables using mathematical operations.
An algebraic expression is formed by combining variables (symbols representing unknown values) and constants using operations like addition, subtraction, multiplication, and division. For example, in the expression 4x + 5y + 3, the parts separated by plus or minus signs are called terms. Here, 4x, 5y, and 3 are the terms.
The numerical factor of a variable term is called its coefficient. In the term 4x, the coefficient of x is 4. In more complex expressions like 200l + 160w + 50lw, the terms are 200l, 160w, and 50lw. The coefficients are 200 (for l), 160 (for w), and 50 (for the product lw). These expressions allow us to model real-world scenarios, such as calculating the total cost of items or the area of a garden.
| Expression | Terms | Variables | Coefficients | Constants |
|---|---|---|---|---|
| 4x + 5y + 3 | 4x, 5y, 3 | x, y | 4, 5 | 3 |
| 200l + 160w + 50lw | 200l, 160w, 50lw | l, w | 200, 160, 50 (for lw) | None |
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Question
In the expression 7z - 4, identify the variable, coefficient, and constant.
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02 · Explore
Univariate Polynomials and Degree
When an algebraic expression involves only one variable, it is called a univariate polynomial.
A univariate polynomial is an expression where all terms contain the same single variable (like x, y, or z) raised to non-negative integer powers. The highest power of the variable in such a polynomial determines its degree. For example, in x² + 5x + 1, the highest power is 2, so the degree is 2.
Polynomials are classified by their degree. A degree 1 polynomial is a linear polynomial (e.g., 3z + 7). Degree 2 is quadratic (e.g., x² + 5x + 1), and degree 3 is cubic (e.g., 5y³ + y² + 2y - 1). A constant number like 8 is a constant polynomial of degree 0 because it can be written as 8x⁰.
Identifying Degree and Coefficients
P(x) = x⁴ - 3x³ + 6x² - 2x + 7
The highest power of x is 4, so the degree is 4. The coefficient of x³ is -3, the coefficient of x² is 6, and the constant term is 7.
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Question
What is the degree of the polynomial 4z - 3?
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03 · Explore
Linear Polynomials and Patterns
Linear polynomials represent relationships where a change in the input leads to a constant change in the output.
A linear polynomial has the general form ax + b. A key characteristic of linear patterns is that the difference between successive values at equal intervals is constant. For example, if a chess club charges a ₹200 joining fee plus ₹50 per match, the total cost for 'm' matches is 200 + 50m. For every additional match, the cost increases by exactly ₹50.
This constant difference defines a linear relationship. Whether it is the perimeter of a square (4x) or a taxi fare, if the rate of change remains the same, the underlying expression is a linear polynomial. In the specific case of a square with perimeter P = 4x, if the side x increases by 0.5 cm, the perimeter increases by 4 * 0.5 = 2 cm.
| Matches (m) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Cost (₹) | 250 | 300 | 350 | 400 |
| Difference | — | 50 | 50 | 50 |
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Question
If a player paid ₹750 for the chess club (200 + 50m), how many matches did they play?
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04 · Explore
Polynomials as Input-Output Processes
Polynomials can be viewed as machines that take an input value and produce a specific output.
When we substitute a specific value for the variable in a polynomial, we calculate the value of the polynomial at that point. This is often called a function. For the linear polynomial 2x + 3, if we provide an input of x = 4, the output is 2(4) + 3 = 11. If x = -6, the output is 2(-6) + 3 = -9.
This process applies to polynomials of any degree. For a quadratic polynomial like 10x - x², if x = 6, the value is 10(6) - (6)² = 60 - 36 = 24. Evaluating these expressions is essential for solving real-world problems involving ages, dimensions, or finances.
The Function Machine
- 1
Input (x)
A value is chosen for the variable, e.g., x = 4.
- 2
Process (2x + 3)
The variable is multiplied by 2 and 3 is added.
- 3
Output (y)
The resulting value is produced, e.g., y = 11.
A linear polynomial acts as a function where every input x results in a unique output y.
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Question
Find the value of 5x - 3 when x = -1.
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