Zeroes of a Polynomial

Finding the values of variables that make the polynomial equal to zero.

What is a Zero of a Polynomial?

A real number 'c' is called a zero of a polynomial p(x) if p(c) = 0.

If p(x) = x - 2, then p(2) = 2 - 2 = 0. So, 2 is a zero of p(x).

A zero of a polynomial is also called a root of the polynomial equation p(x) = 0.

Finding Zeroes of a Linear Polynomial

To find the zero of a linear polynomial p(x) = ax + b, we solve the equation p(x) = 0.
ax + b = 0
ax = -b
x = -b/a

Example: Find zero of p(x) = 2x + 3.
2x + 3 = 0 → 2x = -3 → x = -3/2.

Important Points

  • A non-zero constant polynomial has no zero.
  • Every real number is a zero of the zero polynomial.
  • A linear polynomial has one and only one zero.
  • A polynomial can have more than one zero (equal to its degree).

Practice Questions

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Test your understanding of Zeroes of a Polynomial.

1 Find the zero of the polynomial p(x) = 2x + 5.
  • A 2/5
  • B -5/2
  • C 5/2
  • D -2/5
Explanation:
Set p(x) = 0. 2x + 5 = 0 → 2x = -5 → x = -5/2.
2 How many zeroes can a linear polynomial have?
  • A Exactly one
  • B Exactly two
  • C Infinitely many
  • D Zero or one
Explanation:
A linear polynomial (degree 1) always has exactly one zero.
3 If p(x) = x² - 2x, then p(2) is:
  • A 0
  • B 2
  • C -2
  • D 4
Explanation:
p(2) = (2)² - 2(2) = 4 - 4 = 0. Thus, 2 is a zero of the polynomial.
4 The zero of the zero polynomial is:
  • A 0
  • B 1
  • C Any real number
  • D Not defined
Explanation:
For the zero polynomial p(x) = 0, p(c) = 0 for any real number c. So every real number is a zero.
5 Calculate the zero of p(x) = cx + d.
  • A -d
  • B -c/d
  • C -d/c
  • D d/c
Explanation:
cx + d = 0 → cx = -d → x = -d/c.
6 Check if -1 is a zero of x² - 1.
  • A Yes
  • B No
  • C Only if x > 0
  • D Cannot determine
Explanation:
p(-1) = (-1)² - 1 = 1 - 1 = 0. Since the value is 0, -1 is a zero.
7 Which of the following polynomials has a zero at x = 2?
  • A x + 2
  • B x² + 2
  • C 2x - 4
  • D 2x + 4
Explanation:
For C: p(2) = 2(2) - 4 = 4 - 4 = 0. So x = 2 is a zero.
8 If 1 is a zero of polynomial p(x) = ax² - 3(a-1)x - 1, then value of 'a' is:
  • A 1
  • B -1
  • C 2
  • D -2
Explanation:
p(1) = 0. a(1)² - 3(a-1)(1) - 1 = 0
a - 3a + 3 - 1 = 0
-2a + 2 = 0 → 2a = 2 → a = 1.
9 Identify the polynomial which has 0 and 1 as zeroes.
  • A x² + x
  • B x² - x
  • C x² - 1
  • D x + 1
Explanation:
For B: p(0) = 0² - 0 = 0. p(1) = 1² - 1 = 0. Both are zeroes.
10 A quadratic polynomial can have at most:
  • A 1 zero
  • B 2 zeroes
  • C 3 zeroes
  • D Infinite zeroes
Explanation:
A polynomial of degree 'n' can have at most 'n' zeroes. Quadratic implies degree 2, so at most 2 zeroes.

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Zeroes of a Polynomial - Exam Preparation Strategy

When studying Zeroes of a Polynomial for your final board exams, it is critical to focus on the core concepts and fundamental formulas. Relying strictly on NCERT textbook solutions and practicing previous year questions (PYQs) is the proven methodology for scoring high marks. Avoid rote memorization and instead focus on the logical application of the theories presented in this chapter.

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❓ Frequently Asked Questions

How can I quickly memorize the concepts of Zeroes of a Polynomial?

The most effective way is to create short, handwritten revision notes and continuously test your knowledge using our interactive Mock Tests. Spaced repetition and active recall are much better than passive reading.

What type of questions are most commonly asked from Zeroes of a Polynomial?

Board exams tend to favor conceptual application questions and direct formula-based derivations from the NCERT syllabus. Ensure you have solved every single exercise in the official textbook.

Is reading the NCERT book enough for this chapter?

Yes, the NCERT textbook is the absolute gold standard for board exams. However, to improve your speed and accuracy during the actual exam, you must supplement your reading by solving timed mock tests and objective questions.