Algebraic expressions: terms, coefficients and constants
An algebraic expression mixes numbers and letters with +, −, × and ÷. Example: 4x² − 3x + 7.
- A term is one part joined by + or −. Here the terms are 4x², −3x and 7.
- A variable is a letter whose value can change, like x.
- A coefficient is the number stuck to the variable. The coefficient of x² is 4. The coefficient of x is −3.
- A constant is a term with no variable, like 7.
Like terms have the same letter and the same power (5x and −2x). Only like terms can be added: 5x − 2x = 3x.
What is a polynomial?
A polynomial in x is an expression where every power of x is a whole number: 0, 1, 2, 3 and so on.
- Yes: 3x² + 2x − 1, x³ − 5, 7 (a constant), ½x + 4.
- No: 1/x (this is x to the power −1), √x (power ½), x + 2/x².
We write a polynomial as p(x). To find p(2), put x = 2 in it. If p(x) = x² + 1, then p(2) = 4 + 1 = 5.
By number of terms: monomial (1 term, 5x³), binomial (2 terms, x + 3), trinomial (3 terms, x² + x + 1).
Degree of a polynomial
The degree is the highest power of the variable that has a non-zero coefficient.
- Degree 0: a non-zero constant like 9 (called a constant polynomial).
- Degree 1: linear, like 2x + 3.
- Degree 2: quadratic, like x² − 4x + 1.
- Degree 3: cubic, like x³ + 2x.
Always simplify first. (x² + 3x) − x² = 3x has degree 1, not 2. For two variables, add the powers inside each term: 3x²y³ has degree 5. The zero polynomial 0 has no defined degree.
Linear polynomials: growth and decay models
When something changes by the same amount every step, a linear polynomial describes it.
- Growth: a savings box has ₹50 and you add ₹20 each week: S = 20x + 50. The amount goes up by 20 each week.
- Decay: a candle is 24 cm and burns 3 cm per hour: L = 24 − 3x. It goes down by 3 each hour and is gone at x = 8.
Check a table: if the difference between one value and the next is always the same, the pattern is linear. 5, 8, 11, 14 → difference 3 each time → y = 3x + 5.
Slope and y-intercept of y = ax + b
The graph of y = ax + b is a straight line.
- Slope a = rise ÷ run = change in y when x goes up by 1. a > 0: line goes up (growth). a < 0: line goes down (decay). a = 0: flat line.
- y-intercept b = value of y when x = 0. The line crosses the y-axis at (0, b).
From two points (x₁, y₁) and (x₂, y₂): slope a = (y₂ − y₁) ÷ (x₂ − x₁). Then b = y₁ − a·x₁.
Example: points (1, 5) and (3, 11). a = 6 ÷ 2 = 3. b = 5 − 3 = 2. So y = 3x + 2.
Try it: a practical
Fill a bottle with water to a mark. Every minute, pour out one small cup and measure the height with a ruler. Write the heights in a table. Is the drop the same each minute? Then height = b − ax. Find a (drop per minute) and b (starting height). In the 3D, set the same a and b in free play and compare.
Board exam corner
Common 1–3 mark questions: say whether an expression is a polynomial; find the degree; classify as linear/quadratic/cubic; find p(a); write a linear rule from a word problem; read slope and intercept from y = ax + b or from two points.
Key formulas and definitions
- Polynomial: powers of x are whole numbers 0, 1, 2, …
- Degree = highest power with a non-zero coefficient
- Linear: y = ax + b (a ≠ 0)
- Slope a = (y₂ − y₁) ÷ (x₂ − x₁)
- y-intercept b = value of y at x = 0
- Growth: a > 0 · Decay: a < 0
Worked examples
1. Find the terms, coefficient of x and the constant in 5x² − 7x + 2.
Terms: 5x², −7x, 2. Coefficient of x = −7. Constant = 2.
2. Is x² + 1/x a polynomial?
1/x = x⁻¹. The power −1 is not a whole number, so it is not a polynomial.
3. Find the degree of (x³ + 4x) − (x³ − 2).
Simplify: x³ + 4x − x³ + 2 = 4x + 2. Highest power is 1, so degree 1 (linear).
4. If p(x) = 2x² − 3x + 1, find p(2) and p(−1).
p(2) = 2(4) − 6 + 1 = 3. p(−1) = 2(1) + 3 + 1 = 6.
5. A taxi charges ₹40 plus ₹15 per km. Write the fare as a polynomial and find the fare for 8 km.
Fare F = 15x + 40. For x = 8: F = 120 + 40 = ₹160. Slope 15 = cost per km, intercept 40 = fixed charge.
6. A tank holds 600 L and drains 25 L per minute. When is it empty?
W = 600 − 25x. Empty means W = 0: 25x = 600, x = 24 minutes. This is decay (slope −25).
7. A line passes through (2, 7) and (5, 16). Find its equation y = ax + b.
a = (16 − 7) ÷ (5 − 2) = 9 ÷ 3 = 3. b = 7 − 3 × 2 = 1. So y = 3x + 1.
Common mistakes
- Calling 1/x or √x a polynomial. Their powers (−1, ½) are not whole numbers.
- Finding the degree before simplifying. Cancel terms first, then look for the highest power.
- Forgetting the sign with a coefficient: in 4 − 3x the coefficient of x is −3, not 3.
- Mixing up slope and intercept: in y = 5 + 2x the slope is 2 and the intercept is 5.