Polynomials – Class 10 Mathematics notes, important questions and MCQs

Chapter overview

A polynomial's zeroes are the x-values where it becomes 0 — where its graph meets the x-axis. For quadratics, the zeroes are tied to the coefficients.

Degree and graphs

• Degree 1: linear (ax + b), at most 1 zero; graph is a straight line.
• Degree 2: quadratic (ax² + bx + c), at most 2 zeroes; graph is a parabola.
• Degree 3: cubic, at most 3 zeroes.
• The number of zeroes = number of points where the graph meets the x-axis.

Zeroes and coefficients

• For ax² + bx + c with zeroes α, β: α + β = −b/a and αβ = c/a.
• A quadratic with given sum S and product P of zeroes: k(x² − Sx + P).
• Parabola opens upward if a > 0 and downward if a < 0.

Key terms

Zero of a polynomial

a value of x that makes p(x) = 0

Degree

the highest power of the variable

Parabola

the U-shaped graph of a quadratic polynomial

Formulas

• α + β = −b/a
• αβ = c/a
• Quadratic from zeroes: x² − (α + β)x + αβ

Important questions with answers

Quick revision

• Linear has 1 zero, quadratic at most 2, cubic at most 3.
• Sum of zeroes = −b/a, product = c/a.
• Zeroes are x-intercepts of the graph.

Exam tips

• Count zeroes from a graph only where it touches or cuts the x-axis, not the y-axis.
• Keep the sign of b carefully in α + β = −b/a.

Practice MCQs

1. The degree of 5x³ − 2x + 7 is

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

   Answer: (c) 3. The highest power of x is 3.

2. A quadratic polynomial can have at most how many zeroes?

   1. 1
   2. infinitely many
   3. 3
   4. 2
   Show answer

   Answer: (d) 2. Degree 2 gives at most 2 zeroes.

3. The zero of 2x − 6 is

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

   Answer: (d) 3. 2x − 6 = 0 ⇒ x = 3.

4. Sum of the zeroes of x² − 5x + 6 is

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

   Answer: (d) 5. −b/a = −(−5)/1 = 5.

5. Product of the zeroes of 3x² + 5x − 2 is

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

   Answer: (b) −2/3. c/a = −2/3.

6. A quadratic whose zeroes have sum 4 and product 3 is

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

   Answer: (c) x² − 4x + 3. x² − (sum)x + product = x² − 4x + 3.

7. If one zero of x² − kx + 8 is 2, then k is

   1. 4
   2. −6
   3. 8
   4. 6
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   Answer: (d) 6. 4 − 2k + 8 = 0 ⇒ k = 6.

8. If α, β are zeroes of x² − 7x + 10, then α² + β² is

   1. 69
   2. 49
   3. 20
   4. 29
   Show answer

   Answer: (d) 29. (α + β)² − 2αβ = 49 − 20 = 29.