This MCQ module is based on: Algebraic Expressions, Variables, Terms & Coefficients
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Algebraic Expressions, Variables, Terms & Coefficients
🎓 Class 9
Mathematics
CBSE
Theory
Ch 2 — Introduction to Linear Polynomials
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: Algebraic Expressions, Variables, Terms & Coefficients
Targeting Class 9 level in Algebra, with Intermediate difficulty.
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2.1 Introduction — From Numbers to Symbols
In earlier classes you have worked with numerical expressions like \(3 + 4\) or \(7 \times 5\). Now we step up: an algebraic expression? uses letters such as \(x, y, n, t\) to stand in for numbers we don't yet know — or numbers that change. The letters are called variables?.
Example 1 — Raju's Pencil Boxes
Raju walks into a stationery shop. There are sealed boxes of different colours on sale. The shop owner tells Raju that:
- each red box has 4 pencils,
- each blue box has 5 pencils,
- each green box has 3 extra pencils, free, glued on top of a regular blue box.
If Raju buys \(x\) red boxes, \(y\) blue boxes and \(z\) green boxes, the total number of pencils he gets is:
\(4x + 5y + (5z + 3z) = 4x + 5y + 8z\)
Fig 2.1: Three types of pencil boxes
Key Vocabulary
In the expression \(4x + 5y + 8z\):• \(x, y, z\) are the variables — they can take different numerical values.
• \(4x\), \(5y\), \(8z\) are terms — pieces separated by \(+\) or \(-\) signs.
• \(4, 5, 8\) are the coefficients of \(x, y, z\) respectively — the numerical multipliers.
• A number alone, like 7 in \(2x + 7\), is called a constant.
Example 2 — Rectangular Garden
A rectangular garden has length \(\ell\) metres and width \(b\) metres. A wired fence is to be laid along the entire boundary, costing ₹60 per metre. Special seeds for the lawn are sown across the area and cost ₹50 per square metre.
Length of fence = perimeter = \(2(\ell + b) = 2\ell + 2b\) metres. So fence cost = \(60(2\ell + 2b) = 120\ell + 120b\) rupees.
Area of garden = \(\ell \times b\) sq m. Seed cost = \(50 \ell b\) rupees.
Total cost \(= 120\ell + 120b + 50\ell b\) rupees.
Think and Reflect: Identify the terms and coefficients of \(120\ell + 120b + 50\ell b\). Are the first two expressions different from the third one in any structural way?
Terms: \(120\ell\), \(120b\), \(50\ell b\). Coefficients: 120, 120, 50. The first two terms are linear (single-variable, power 1) but \(50\ell b\) involves a product of two variables, raising the overall degree.
Example 3 — Two Rectangular Plots
Suppose two adjacent rectangular plots are to be sodded. The first has dimensions \((10 - x)\) m by 3 m, the second \((5+x)\) m by 4 m. The cost of grass is ₹100 per m².
Area = \(3(10 - x) + 4(5 + x) = 30 - 3x + 20 + 4x = 50 + x\) sq m.
Cost = \(100(50 + x) = 5000 + 100x\) rupees.
Think and Reflect: Identify the terms and coefficients of \(5000 + 100x\). Notice anything common between Example 1 and Example 3?
Both Examples 1 and 3 involve only first-power variables — no \(x^2\), no \(xy\). They are linear. Example 2's term \(50\ell b\) makes it non-linear.
2.1.1 Variables, Terms, and Coefficients — Formal View
| Expression | Variables | Terms | Coefficients | Constant |
|---|---|---|---|---|
| \(4x + 5y + 8z\) | x, y, z | 4x, 5y, 8z | 4, 5, 8 | 0 |
| \(2y^2 - 3y + 7\) | y | \(2y^2, -3y, 7\) | 2, −3 | 7 |
| \(\frac{1}{2}x - \sqrt{2}\) | x | \(\frac{1}{2}x, -\sqrt{2}\) | \(\tfrac{1}{2}\) | \(-\sqrt{2}\) |
| \(5x^2 + 6x - 1\) | x | \(5x^2, 6x, -1\) | 5, 6 | −1 |
Exercise Set 2.1
Q1. Find the degree of the following expressions:
(i) \(2x^3 - 5x\) (ii) \(y^2 - 7y + 1\) (iii) 11 (iv) \(x^2 y\)
(i) \(2x^3 - 5x\) (ii) \(y^2 - 7y + 1\) (iii) 11 (iv) \(x^2 y\)
(i) Degree 3 (highest power of x). (ii) Degree 2. (iii) Degree 0 (a constant). (iv) Degree \(2 + 1 = 3\) (sum of exponents in the term).
Q2. Write polynomials of degree 1, 2 and 3 (one each).
Sample answers — Degree 1: \(3x - 7\). Degree 2: \(x^2 + 4x - 5\). Degree 3: \(2x^3 - x + 1\). (Many valid answers.)
Q3. What are the coefficients of \(x^2\) and \(x^3\) in the expression \(x^4 - 3x^3 + 6x^2 - 2x + 7\)?
Coefficient of \(x^3\) is \(-3\); coefficient of \(x^2\) is \(6\).
Q4. What is the coefficient of \(z\) in \(4z^2 + 5z - 1\)?
The coefficient of \(z\) (i.e., \(z^1\)) is 5.
Q5. What is the constant term of degree 0 polynomial \(9x^2 + 5x - 8x - 10\)? In this chapter, we shall study linear polynomials.
Combining like terms: \(9x^2 + (5-8)x - 10 = 9x^2 - 3x - 10\). The constant term is −10.
Activity: Build-an-Expression Game
L3 ApplyMaterials: Index cards, dice
Predict: How many distinct expressions can two players build using only \(x\), 2, +, and one term?
- Make 12 cards with letters \(x, y, z\) (4 each), 12 cards with digits 0–9.
- Each player draws 3 letter cards and 3 digit cards.
- Build an algebraic expression with at least 3 terms using only your cards and the operations +, −, ×.
- Identify variables, terms, coefficients and the constant. Score 1 point for each correctly identified element.
- Highest score after 5 rounds wins.
Use repeated variables to create like-terms (e.g. \(3x + 5x = 8x\)) for fast identification. Mixing letter pairs (\(xy\)) lets you raise degree quickly without using exponents.
Competency-Based Questions
Scenario: A school stationery shop sells notebooks at ₹40 each, pens at ₹15 each, and a fixed ₹20 carry-bag charge. A student buys \(n\) notebooks and \(p\) pens.
Q1. Write an algebraic expression for the student's total bill.
L3 ApplyTotal bill \(= 40n + 15p + 20\) rupees.
Q2. Identify variables, terms, coefficients, and constant in your expression.
L4 AnalyseVariables: \(n, p\). Terms: \(40n, 15p, 20\). Coefficients: 40, 15. Constant: 20.
Q3. A friend claims that adding more pens always raises the cost faster than adding notebooks "because pens have more variety." Evaluate.
L5 EvaluateWrong. The rate of cost change per item equals the coefficient. Coefficient of \(n\) is 40, of \(p\) is 15. Each extra notebook raises the cost by ₹40 — much faster than each extra pen (₹15). The friend is mathematically incorrect.
Q4. Design a stationery deal where the total bill expression is \(50n + 10p\) (no constant), and explain how the shop could offer this in real life.
L6 CreateNotebooks at ₹50 each, pens at ₹10 each, with no carry-bag charge. The shop could waive the bag charge for purchases inside the store, or include it in the per-item price already.
Assertion–Reason Questions
A: In \(4x + 5y + 8z\), the coefficient of \(z\) is 8.
R: The coefficient of a variable is the numerical factor multiplying it.
R: The coefficient of a variable is the numerical factor multiplying it.
Answer: (a)
A: The expression \(7\) is an algebraic expression of degree 0.
R: A constant can be written as \(7 \cdot x^0\), and the highest power of \(x\) is 0.
R: A constant can be written as \(7 \cdot x^0\), and the highest power of \(x\) is 0.
Answer: (a) — A non-zero constant has degree 0 by exactly this reasoning.
A: \(50\ell b\) is a linear term.
R: A term is linear iff each variable in it appears to the first power and no higher.
R: A term is linear iff each variable in it appears to the first power and no higher.
Answer: (d) — A is false: \(50\ell b\) has total degree 2 (it's a product of two variables) so it is not linear. R is true (correct general rule about linear terms in a single variable; for multivariate, "linear" requires sum of exponents \(\le 1\)).
Frequently Asked Questions
What is an algebraic expression in Class 9 Maths?
An algebraic expression combines variables (letters representing numbers) and constants using operations like addition, subtraction and multiplication. For example, 4x + 5y + 8z is an algebraic expression with three terms.
What is the difference between a variable and a constant?
A variable is a symbol such as x, y or n that can take different numerical values. A constant is a fixed number that does not change, such as 3, -7 or pi. In 4x + 5, x is the variable and 4 and 5 are constants.
What are terms and coefficients of an algebraic expression?
A term is each part of the expression separated by + or - signs. The coefficient is the numerical factor in a term. In 4x + 5y + 8z, the terms are 4x, 5y and 8z, and the coefficients of x, y and z are 4, 5 and 8 respectively.
What are like and unlike terms?
Like terms have exactly the same variable part (same letters with the same exponents); only their coefficients may differ, e.g. 3x and 7x. Unlike terms have different variable parts, such as 3x and 5y, and cannot be combined by simple addition.
How do you translate a word problem into an algebraic expression?
Identify the unknown quantities and assign letters (variables) to them, then express each given relation in symbols. For example, if a red box has 4 pencils and Raju buys x red boxes, the number of pencils is 4x; for x red, y blue and z green boxes the total is 4x + 5y + 8z.
Why do we use letters instead of numbers in algebra?
Letters let us write a single rule that works for every value of the unknown. Instead of repeating the same calculation for each side length, the formula 4x gives the perimeter of a square for any positive x.
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