Introduction to Linear Polynomials Class 9: Notes and Questions

This guide to Introduction to Linear Polynomials Class 9 walks you through the whole chapter in one read: terms and coefficients, degree, linear patterns, growth and decay, finding y = ax + b from two values, and reading slope and y-intercept on a graph. Every idea comes with a worked example, and the important questions at the end have full model answers.

Algebraic expressions: terms, coefficients and constants

An algebraic expression combines variables (symbols for unknown values) and constants using operations such as addition, subtraction, multiplication and division.

• Terms are the parts separated by plus or minus signs. In 4x + 5y + 3, the terms are 4x, 5y and 3.
• Coefficient is the numerical factor of a variable term. In 4x, the coefficient of x is 4.
• Constant is the term with no variable. In 4x + 5y + 3, the constant is 3.

Notice that in 200l + 160w + 50lw, the coefficient 50 belongs to the product lw, not to l or w alone. Expressions like these help us model real situations, such as the total cost of items or the area of a garden.

Quick example: In 7z - 4, the variable is z, the coefficient of z is 7, and the constant term is -4. Keep the minus sign with the constant.

Univariate polynomials and their degree

A univariate polynomial uses only one variable (such as x, y or z), and every power of that variable is a non-negative integer. The degree is the highest power of the variable.

Worked example: For 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.

For 4z - 3, the highest power of z is 1, so it is a linear polynomial of degree 1.

Linear polynomials and constant differences

A linear polynomial has the general form ax + b. Its key feature: when the input goes up in equal steps, the output changes by the same amount every time.

Take a chess club that charges a ₹200 joining fee plus ₹50 per match. The cost for m matches is 200 + 50m.

The difference stays at 50, so the relationship is linear. The perimeter of a square, P = 4x, works the same way: if the side x increases by 0.5 cm, the perimeter increases by 4 × 0.5 = 2 cm.

Working backwards: If a player paid ₹750, then 200 + 50m = 750. Subtract 200 to get 50m = 550. Divide by 50 to get m = 11 matches.

Polynomials as input-output machines

Think of a polynomial as a machine. You put in a value of the variable, the machine follows the rule, and a value comes out. Finding this output is called evaluating the polynomial at that value.

• Input: choose x = 4.
• Process (2x + 3): multiply by 2, then add 3.
• Output: 2(4) + 3 = 11.

With x = -6, the same machine gives 2(-6) + 3 = -12 + 3 = -9.

This works for any degree. For the quadratic 10x - x² at x = 6: 10(6) - (6)² = 60 - 36 = 24.

One more: 5x - 3 at x = -1 gives 5(-1) - 3 = -5 - 3 = -8. Put negative inputs inside brackets so you do not lose the sign.

Linear growth and linear decay

• Linear growth: a quantity increases by a constant amount over equal intervals. A plant 1.75 feet tall that grows 0.5 feet every month has height h = 1.75 + 0.5t after t months. After 4 months, h = 1.75 + 0.5(4) = 1.75 + 2 = 3.75 feet.
• Linear decay: a quantity decreases by a constant amount over equal intervals. A tank with 3 m of water that drops by 0.5 m each month has h = 3 - 0.5t. After 2 months, h = 3 - 1 = 2 m.

Exercise-style example: A phone bought for ₹10,000 loses ₹800 in value each year, so v(t) = 10000 - 800t. After 3 years, v(3) = 10000 - 800(3) = 10000 - 2400 = ₹7,600.

The expression b(x) = 600 - 15x shows decay, because the value falls by 15 for every unit increase in x.

Finding y = ax + b from two pairs of values

If you know two related pairs (x₁, y₁) and (x₂, y₂), you can find a and b.

1. Pick out the two (x, y) pairs from the problem.
2. Substitute both into y = ax + b.
3. Subtract one equation from the other to find a (the rate of change).
4. Put a back into either equation to find b (the starting value).
5. Write the final equation.

Worked example: A bill is ₹350 for 10 GB and ₹550 for 20 GB.

• 350 = 10a + b and 550 = 20a + b
• Subtract: 200 = 10a, so a = 20
• Then 350 = 10(20) + b = 200 + b, so b = 150
• Relationship: y = 20x + 150. Here 150 is the fixed fee and 20 is the cost per GB.

Check: 20(20) + 150 = 400 + 150 = 550. Correct.

Graphs, slope and y-intercept

To draw y = 2x + 1, you need at least two points. Take x = 0 (y = 1) and x = 3 (y = 7), plot (0, 1) and (3, 7), and join them with a ruler. Every point on the line satisfies the equation: for (7, 15), 2(7) + 1 = 15, so it lies on the line. If your points do not form a straight line, re-check your arithmetic.

In y = ax + b:

• a is the slope (steepness). If a > 0 the line rises (growth); if a < 0 it falls (decay). If a = 1, the line is equally inclined to both axes. A larger a means a steeper line.
• b is the y-intercept, where the line crosses the y-axis at (0, b). Changing b while keeping a the same shifts the line up or down, parallel to the original.

Lines with the same slope are always parallel.

Remember this

• Terms are separated by + and - signs; the sign belongs to the term.
• Degree = highest power of the variable. A constant like 8 has degree 0.
• Linear means equal steps in input give equal changes in output.
• Growth: + constant amount each interval. Decay: - constant amount each interval.
• In y = ax + b, a is the slope and b is the y-intercept at (0, b).
• Same slope means parallel lines.

Important questions with answers

1. Write the terms and coefficients of 200l + 160w + 50lw. Terms: 200l, 160w and 50lw. Coefficients: 200 for l, 160 for w, 50 for lw. There is no constant term.

2. Find the degree of 5y³ + y² + 2y - 1 and name the polynomial. The highest power of y is 3, so the degree is 3. It is a cubic polynomial.

3. Why is 8 called a polynomial of degree 0? Because 8 can be written as 8x⁰. The power of x is 0, so the degree is 0.

4. Find the value of 10x - x² when x = 2. 10(2) - (2)² = 20 - 4 = 16.

5. A chess club charges 200 + 50m for m matches. How many matches cost ₹750? 200 + 50m = 750, so 50m = 550 and m = 11.

6. Is b(x) = 600 - 15x growth or decay? Explain. Decay. The value goes down by a fixed 15 for every unit increase in x.

7. If y = ax + b passes through (1, 5) and (3, 11), find a and b. 5 = a + b and 11 = 3a + b. Subtracting gives 6 = 2a, so a = 3. Then b = 5 - 3 = 2. The equation is y = 3x + 2. Check: 3(3) + 2 = 11.

8. Does the point (4, 10) lie on y = 2x + 1? 2(4) + 1 = 9, which is not 10. So (4, 10) does not lie on the line.

9. Where does y = 3x - 2 cut the y-axis? The y-intercept is -2, so the line cuts the y-axis at (0, -2).

10. Are y = 4x + 7 and y = 4x - 3 parallel? Yes. Both have slope 4 but different y-intercepts, so they are parallel.

Common mistakes to avoid

• Dropping the minus sign: in 7z - 4, the constant is -4, not 4.
• Mixing up coefficient and power: in 5y³, 5 is the coefficient and 3 is the power.
• Forgetting brackets for negative inputs: 2(-6) + 3, not 2 - 6 + 3.
• Swapping slope and intercept: in y = ax + b, the number multiplying x is the slope.
• Drawing a line from only one point, or joining points freehand. Use two or more points and a ruler.

When you are ready to practise these steps with guidance, study this chapter with Joy on Learnijoy.