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CBSE · Class 9 · Mathematics

Chapter 2: Linear Polynomials

5 concepts · 40 practice questions with answers · Free to read

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Linear polynomials: terms and values

From daily life

A taxi charges ₹20 to start plus ₹12 per km. The bill for x km is 12x + 20. Put in a number for x and you get the bill.

Simple definition

A linear polynomial in x has the form ax + b (a ≠ 0). x is the variable, a is the coefficient and b is the constant term. Its value at x = k is ak + b.

Formula: p(x) = ax + b, p(k) = ak + b

3x+2term with xconstantx = 43(4)+2 = 14

In the figure: For p(x) = 3x + 2: coefficient 3, constant 2, and p(4) = 3 × 4 + 2 = 14.

Rule 1: Find the value — Replace x by the number and calculate; mind the signs.

  • Easy: p(x) = 2x + 5: p(3) = 11.
  • Medium: p(x) = 4x − 7: p(−2) = −8 − 7 = −15.
  • Exam-type: p(x) = 5 − 3x: p(2) = 5 − 6 = −1.

Rule 2: Real-life rule — Fixed part is the constant; the part that changes with x carries the coefficient.

  • Easy: ₹20 + ₹12 per km: p(x) = 12x + 20; for 5 km ₹80.
  • Medium: Pencils at ₹5 each: cost = 5x.
  • Exam-type: A pack of 10 sweets plus x more: total = x + 10.
Remember

A linear polynomial has x only to the power 1. x² or 1/x makes it non-linear.

Common mistake

Writing p(−2) for 4x − 7 as 4 × 2 − 7 and forgetting the minus sign.

Memory clue

"Bracket first": put the number in brackets before multiplying.

Practice questions

Easy
  1. 1Pick the expression that is a polynomial of degree 1 in y.
  2. 2What is the constant term of the polynomial −8t − 6?
  3. 3Evaluate p(x) = −7x − 3 at x = −3.
Medium
  1. 4For the polynomial p(x) = 4x + b, it is known that p(3) = 9. Find b.
  2. 5If p(x) is a linear polynomial with p(1) = −1 and p(6) = −21, find p(8).
  3. 6If p(x) = −6x + 4 and q(x) = −6x + 4, find p(x) + q(x).
Challenging
  1. 7A linear polynomial p(x) has p(1) = −4 and p(−1) = −10. Find p(0).
  2. 8Let p(x) = 4x − 3. Find p(p(2)).
Show answers (try them yourself first)
  1. 13y + 5
  2. 2−6
  3. 318
  4. 4−3
  5. 5−29
  6. 6−12x + 8
  7. 7−7
  8. 817

Linear patterns and the n-th term

From daily life

Make squares from matchsticks in a row: 1 square needs 4 sticks, 2 need 7, 3 need 10. Each new square adds 3 sticks. A rule gives the sticks for any number of squares.

Simple definition

If a pattern grows by the same amount d each time, its n-th term is dn + (first term − d), a linear polynomial in n. The step d is the coefficient of n.

Formula: n-th term = dn + (a − d), where a = first term

581114+3+3+3n-th term = 3n + 2

In the figure: The pattern 5, 8, 11, 14 adds 3 each time, so the n-th term is 3n + 2 (n = 1 gives 5).

Rule 1: Find the rule — Subtract consecutive terms to get d, then use dn + (a − d).

  • Easy: 2, 5, 8, …: d = 3, rule 3n − 1.
  • Medium: 10, 20, 30, …: rule 10n.
  • Exam-type: 7, 5, 3, …: d = −2, rule −2n + 9.

Rule 2: Use the rule — Put the value of n into the rule, or solve for n when a term is given.

  • Easy: Rule 3n + 2: 10th term = 32.
  • Medium: Rule 4n − 1 = 39 gives n = 10.
  • Exam-type: Sticks for 6 squares: 3 × 6 + 1 = 19.
Remember

Always test the rule on the first two terms before using it.

Common mistake

Writing the rule as dn only and forgetting the (a − d) part.

Memory clue

"Step times n, then fix the start."

Practice questions

Easy
  1. 1In the pattern 7, 1, −5, −11, … what is the constant difference between consecutive terms?
  2. 2The n-th term of a pattern is T(n) = 5n + 6. Find T(13).
  3. 3A pattern has first term 12 and each next term is 5 more than the previous one. Which rule gives the n-th term?
Medium
  1. 4For which value of n does the rule 3n + 8 give 56?
  2. 5A pattern of shapes uses 6 sticks for the first shape, and each next shape needs 3 more sticks than the one before. How many sticks are needed for shape number 11?
  3. 6A pattern of shapes uses 3 sticks for the first shape, and each next shape needs 2 more sticks than the one before. How many sticks are needed for shape number 10?
Challenging
  1. 7A linear pattern has T(2) = 12 and T(6) = 36. Find T(12).
  2. 8The n-th term of a pattern is 7n − 4. By how much does the 14-th term exceed the 6-th term?
Show answers (try them yourself first)
  1. 1−6
  2. 271
  3. 35n + 7
  4. 416
  5. 536
  6. 621
  7. 772
  8. 856

Linear growth and decay

From daily life

A plant 10 cm tall grows 2 cm every week. A tank with 100 L loses 5 L every minute. One grows steadily, the other empties steadily.

Simple definition

When a quantity starts at b and changes by the same rate a each unit of time, its value is a·t + b. Positive a means growth, negative a means decay.

Formula: Value = (rate) × time + (start)

10 + 2wgrowth100 − 5tdecay

In the figure: Green line: 10 + 2w rises 2 per week. Red line: 100 − 5t falls 5 per minute.

Rule 1: Build the rule — Start value is the constant; rate per unit is the coefficient (negative for loss).

  • Easy: ₹500 saved, +₹50 per month: 50m + 500.
  • Medium: Candle 30 cm, burns 2 cm per hour: 30 − 2h.
  • Exam-type: Phone at 80%, falls 4% per hour: 80 − 4h.

Rule 2: Answer questions — Substitute the time to find the value, or set the value and solve for time.

  • Easy: 10 + 2w after 6 weeks: 22 cm.
  • Medium: 100 − 5t = 0 gives t = 20 minutes.
  • Exam-type: When does 30 − 2h = 10? h = 10 hours.
Remember

Same change per unit time means linear. If the change itself keeps changing, it is not linear.

Common mistake

Using the rate as the start value, e.g. writing 2w + 10 as 10w + 2.

Memory clue

"Start stays, rate travels with the variable."

Practice questions

Easy
  1. 1Bhavna's plant is 9 cm tall today and grows 3 cm every week. How tall will it be after 3 weeks?
  2. 2A candle is 36 cm long. It burns 4 cm every hour. What is its length after 2 hours?
  3. 3A cab charges a fixed ₹27 plus ₹17 for every kilometre. What is the fare for 9 km?
Medium
  1. 4A tank holds 476 litres and leaks 17 litres every hour. After how many hours will exactly 306 litres be left?
  2. 5A sapling is 19 cm tall and grows 3 cm every week. In how many weeks will it be 43 cm tall?
  3. 6A bike costs ₹76 thousand today and its resale value drops by ₹7 thousand every year. Which formula gives the value V after t years (in ₹ thousand)?
Challenging
  1. 7Anil's sapling is 18 cm tall and grows 3 cm a week. Firoz's sapling is 26 cm tall and grows 2 cm a week. After how many weeks are both equally tall?
  2. 8Plan A costs ₹187 per month plus ₹13 per GB. Plan B costs ₹43 per month plus ₹25 per GB. For how many GB do both plans cost the same?
Show answers (try them yourself first)
  1. 118
  2. 228
  3. 3180
  4. 410
  5. 58
  6. 6V = 76 − 7t
  7. 78
  8. 812

Zeros of a linear polynomial

From daily life

A shopkeeper asks: for how many items does the bill 2x − 6 become ₹0 after the discount? The answer is the number that makes the expression zero.

Simple definition

A zero of p(x) is a number k with p(k) = 0. A linear polynomial ax + b has exactly one zero, x = −b/a.

Formula: ax + b = 0 ⇒ x = −b/a

0123456zero x = 32(3) − 6 = 0

In the figure: For 2x − 6: a = 2, b = −6, so the zero is −(−6)/2 = 3, and 2(3) − 6 = 0.

Rule 1: Find the zero — Set the polynomial equal to 0 and solve for x.

  • Easy: 3x + 9: zero −3.
  • Medium: 5x − 2: zero 2/5.
  • Exam-type: 7 − x: zero 7.

Rule 2: Check a zero — Put the number in; if the value is 0 it is a zero.

  • Easy: Is 4 a zero of 2x − 8? 2(4) − 8 = 0, yes.
  • Medium: Is 1 a zero of 3x + 1? Value 4, no.
  • Exam-type: If k is a zero of ax + b with k = 2 and b = −10, then a = 5.
Remember

The zero is where the line crosses the x-axis. A linear polynomial never has two zeros.

Common mistake

Giving the zero as b/a and missing the minus sign.

Memory clue

"Move the constant, then divide by the coefficient."

Practice questions

Easy
  1. 1How many zeros does a linear polynomial ax + b (a ≠ 0) have?
  2. 2Find the zero of 5x − 35.
  3. 3Find the zero of the polynomial 2x + 6.
Medium
  1. 4If x = 4 is a zero of the polynomial 7x + b, find b.
  2. 5Solve 6x + 9 = 2x − 8 for x.
  3. 6A block of ice at the temperature 25 °C is cooled so that its temperature falls by 5 °C every minute. After how many minutes does the temperature reach 0 °C?
Challenging
  1. 7p(x) = 4x + 3 and q(x) = 4x + 1. Find the zero of p(x) + q(x).
  2. 8A linear polynomial p(x) has zero x = 3 and p(0) = −12. Find p(5).
Show answers (try them yourself first)
  1. 1exactly one
  2. 27
  3. 3−3
  4. 4−28
  5. 5−17/4
  6. 65
  7. 7−1/2
  8. 88

Graphs of linear relationships

From daily life

A taxi bill plotted against distance makes a straight line. It starts at ₹20 and rises steadily.

Simple definition

The graph of y = ax + b is a straight line. b is where it cuts the y-axis (intercept); a decides the tilt: rising if a > 0, falling if a < 0. The zero is where it meets the x-axis.

Formula: y = ax + b: y-intercept b, zero −b/a

xy−3−2−1123−3−2−112345(0,1)(1,3)

In the figure: The line y = 2x + 1 cuts the y-axis at 1 and passes through (1, 3). Its zero is −1/2.

Rule 1: Plot two points — Any two points are enough; join them and extend the line.

  • Easy: y = x + 2: (0, 2) and (1, 3).
  • Medium: y = 2x: (0, 0) and (2, 4).
  • Exam-type: y = 3 − x: (0, 3) and (3, 0).

Rule 2: Read the graph — Find b where x = 0, and the zero where y = 0.

  • Easy: Line through (0, 4) and (2, 0): y = 4 − 2x.
  • Medium: Falling line: a is negative.
  • Exam-type: Horizontal line y = 3 has a = 0 (not a linear polynomial in x).
Remember

Two points fix a straight line. Check with a third point.

Common mistake

Joining points with a curve or a broken line.

Memory clue

"b is where it begins on the y-axis, a is how steep."

Practice questions

Easy
  1. 1At which point does the graph of y = 2x + 2 cut the y-axis?
  2. 2A point on the line y = x + 7 has x-coordinate −3. What is its y-coordinate?
  3. 3Find the x-coordinate of the point where y = 5x + 5 meets the x-axis.
Medium
  1. 4A straight line passes through (3, 2) and (7, 10). By how much does y change when x increases by 1?
  2. 5A straight line passes through (3, 2) and (5, 6). By how much does y change when x increases by 1?
  3. 6A line passes through (3, 2) and (5, 4). Which is its equation?
Challenging
  1. 7The lines y = 2x + 1 and y = 5x − 5 meet at a point. What is its x-coordinate?
  2. 8The line y = x − 3 together with the two coordinate axes encloses a triangle. Find its area (in square units).
Show answers (try them yourself first)
  1. 1(0, 2)
  2. 24
  3. 3−1
  4. 42
  5. 52
  6. 6y = x − 1
  7. 72
  8. 89/2