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

Free Preview - 10 Questions

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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