Introduction to Linear Polynomials – Class 9 Mathematics notes, important questions and MCQs

Chapter overview

A linear polynomial has degree 1, like ax + b (a ≠ 0). Its graph is a straight line and it has exactly one zero, x = −b/a.

Polynomials

• A polynomial has terms with whole-number powers of the variable.
• Degree = highest power: constant (0), linear (1), quadratic (2), cubic (3).
• Terms: monomial (1 term), binomial (2), trinomial (3).

Linear polynomials

• General form ax + b with a ≠ 0.
• Zero: the value of x where ax + b = 0, i.e. x = −b/a.
• Graph of y = ax + b is a straight line crossing the x-axis at the zero.

Value and remainder

• p(k) means put x = k in p(x).
• Remainder theorem: when p(x) is divided by (x − a), the remainder is p(a).
• (x − a) is a factor of p(x) if p(a) = 0.

Key terms

Coefficient

the number multiplying a variable term

Zero of a polynomial

a value of x making p(x) = 0

Linear polynomial

a polynomial of degree 1

Formulas

• Zero of ax + b is x = −b/a
• Remainder on dividing p(x) by (x − a) = p(a)

Important questions with answers

Quick revision

• Degree 1 → linear → one zero.
• Graph of a linear polynomial is a straight line.
• p(a) = 0 ⇔ (x − a) is a factor.

Exam tips

• x + 1/x or √x are not polynomials (powers must be whole numbers).
• A non-zero constant polynomial has no zero.

Practice MCQs

1. The degree of 7x + 3 is

   1. 0
   2. 1
   3. 2
   4. 7
   Show answer

   Answer: (b) 1. Highest power of x is 1.

2. The zero of x − 5 is

   1. −5
   2. 0
   3. 5
   4. 1/5
   Show answer

   Answer: (c) 5. x − 5 = 0 ⇒ x = 5.

3. Which is NOT a polynomial?

   1. √x + 1
   2. x² − 1
   3. 3x + 2
   4. 5
   Show answer

   Answer: (a) √x + 1. Power 1/2 is not a whole number.

4. The zero of 3x + 6 is

   1. 2
   2. −3
   3. 6
   4. −2
   Show answer

   Answer: (d) −2. x = −6/3 = −2.

5. If p(x) = 2x − 4, then p(3) =

   1. 10
   2. 2
   3. −2
   4. 6
   Show answer

   Answer: (b) 2. 6 − 4 = 2.

6. The remainder when x² + 3x + 1 is divided by (x − 1) is

   1. 1
   2. 5
   3. 3
   4. 0
   Show answer

   Answer: (b) 5. p(1) = 1 + 3 + 1 = 5.

7. For what k is (x − 2) a factor of x² − kx + 6?

   1. 3
   2. 6
   3. 5
   4. 2
   Show answer

   Answer: (c) 5. 4 − 2k + 6 = 0 ⇒ k = 5.

8. The graph of y = 2x − 6 meets the x-axis at

   1. (0, −6)
   2. (3, 0)
   3. (−3, 0)
   4. (6, 0)
   Show answer

   Answer: (b) (3, 0). y = 0 ⇒ x = 3.