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Zeroes of Polynomial | Algebra - Mathematics PDF Download

We already know that a polynomial is an algebraic term with one or many terms. Zeroes of Polynomial are the real values of the variable for which the value of the polynomial becomes zero. So, real numbers, ‘m’ and ‘n’ are zeroes of polynomial p(x), if p(m) = 0 and p(n) = 0.

Understanding Zeroes of Polynomial
Zeroes of Polynomial | Algebra - Mathematics

Example 1

Let’s look at the polynomial, p(x) = 5x3 – 2x2 + 3x -2. Now, let’s find the value of the polynomial(x) at x = 1, p(1) = 5(1)3 – 2(1)2 + 3(1) – 2 = 5 – 2 + 3 – 2 = 4. Therefore, we can say that the value of the polynomial p(x) at x = 1 is 4.
Next, let’s find the value of the polynomial(x) at x = 0, p(0) = 5(0)3 – 2(0)2 + 3(0) – 2 = 0 – 0 + 0 – 2 = – 2. Therefore, we can say that the value of the polynomial p(x) at x = 0 is – 2.
Example 2

Let’s look at another polynomial now, p(x) = x – 1. Let’s find the value of the polynomial at x = 1, p(1) = 1 – 1 = 0. So, the value of the polynomial p(x) at x = 1 is 0. Therefore, 1 is the zero of the polynomial p(x).
Similarly, for the polynomial q(x) = x – 2, the zero of the polynomial is 2. To summarize, zeroes of polynomial p(x) are numbers c and d such that p(c) = 0 and p(d) = 0.
Learn various types of polynomials here in detail.

Calculating Zeroes of polynomial
When we calculated zeroes of polynomial p(x) = x – 1, we equated it to 0, x – 1 = 0 or, x = 1. Hence, we say that p(x) = 0 is the Polynomial Equation. 1 is the Root of the Polynomial equation p(x) = 0. OR 1 is the Zero of the Polynomial equation p(x) = x – 1 = 0.
Now, let’s look at a constant polynomial ‘5’. You can write this as 5x0. What is the Root of this constant polynomial? The answer is a Non-zero constant polynomial has no zero. Also, every real number is a zero of the Zero Polynomial.
Let’s look at the following linear polynomial to understand the calculation of the roots or ‘zeroes of polynomial’: p(x) = ax + b … where a ≠ 0. To find a zero, we must equate the polynomial to 0. [p(x) = 0]. Hence, ax + b = 0 … where a ≠ 0. So, ax = – b or, x = – b/a. Therefore, ‘- b/a’ is the only zero of p(x) = ax + b. We can also say that a linear polynomial has only one zero.

Observations

  • A zero of a polynomial need not be 0.
  • 0 may be a zero of a polynomial.
  • Every linear polynomial has one and only one zero.
  • A polynomial can have more than one zero.

Learn Degree of Polynomials in detail.

More Solved Examples for You
Question: Find p(0), p(1) and p(2) for each of the following polynomials:

  • p(y) = y2 – y + 1
  • p(t) = 2 + t + 2t2 – t3

  • p(x) = x3

  • p(x) = (x – 1) (x + 1)

Solution:

p(y) = y2 – y + 1

  • p(0) = 02 – 0 + 1 = 1
  • p(1) = 12 – 1 + 1 = 1
  • p(2) = 22 – 2 + 1 = 3

p(t) = 2 + t + 2t2 – t3

  • p(0) = 2 + 0 + 2(0)2 – (0)3 = 2 + 0 + 0 – 0 = 2
  • p(1) = 2 + 1 + 2(1)2 – (1)3 = 2 + 1 + 2 – 1 = 4
  • p(2) = 2 + 2 + 2(2)2 – (2)3 = 2 + 2 + 8 – 8 = 4

p(x) = x3

  1. p(0) = (0)3 = 0
  2. p(1) = (1)3 = 1
  3. p(2) = (2)3 = 8

Verify the following

Question: Verify whether the following are zeroes of polynomials indicated against them.

  • p(x) = 3x + 1, x = – 1/3
  • p(x) = 5x – π, x = 4/5
  • p(x) = (x + 1) (x – 2), x = – 1, 2
  • p(x) = 3x2 – 1, x = – 1/√3 , 2/√3

Solution:

  • p(x) = 3x + 1, x = – 1/3

    p(-1/3) = 3(-1/3) + 1 = – 1 + 1 = 0.

    Hence, x = -1/3 is a zero of polynomial 3x + 1.

  • p(x) = 5x – π, x = 4/5

    p(4/5) = 5(4/5) – π = 4 – π  ≠ 0.

    Hence, x = 4/5 is not a zero of polynomial 5x – π.

  • p(x) = (x + 1) (x – 2), x = – 1, 2

    p(-1) = (– 1 + 1)(- 1 – 2) = (0)(- 3) = 0.

    Hence, x = – 1 is a zero of polynomial (x + 1) (x – 2).

    p(2) = (2 + 1)(2 – 2) = (3)(0) = 0.

    Hence, x = 2 is a zero of polynomial (x + 1) (x – 2).

  • p(x) = 3x2 – 1, x = – 1/√3 , 2/√3

    p(- 1/√3) = 3(- 1/√3)2 – 1 = 3 (1/3) – 1 = 1 – 1 = 0.

    Hence, x = – 1/√3 is a zero of polynomial 3x2 – 1.

    p(2/√3) = 3(2/√3) – 1 = 3(4/3) – 1 = 4 – 1 = 3 ≠ 0.

    Hence, x = 2/√3 is not a zero of polynomial 3x2 – 1.

The document Zeroes of Polynomial | Algebra - Mathematics is a part of the Mathematics Course Algebra.
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FAQs on Zeroes of Polynomial - Algebra - Mathematics

1. What are the zeroes of a polynomial?
Ans. The zeroes of a polynomial are the values of the variable that make the polynomial equal to zero. In other words, they are the solutions to the equation formed by setting the polynomial equal to zero.
2. How can I find the zeroes of a polynomial?
Ans. To find the zeroes of a polynomial, you can use various methods such as factoring, synthetic division, or using the quadratic formula for higher degree polynomials. These methods help you solve the polynomial equation and determine the values of the variable for which the polynomial equals zero.
3. Can a polynomial have more than one zero?
Ans. Yes, a polynomial can have more than one zero. In fact, the number of zeroes of a polynomial is equal to its degree. However, some of the zeroes can be repeated, meaning that they have a multiplicity greater than one. This means that a polynomial of degree 'n' can have at most 'n' distinct zeroes.
4. How do I determine the multiplicity of a zero in a polynomial?
Ans. The multiplicity of a zero in a polynomial is determined by how many times the zero appears as a factor in the polynomial's factored form. For example, if a zero appears as a factor twice, its multiplicity is two. To determine the multiplicity, you can factor the polynomial and observe the power of each factor contributing to the zero.
5. Can a polynomial have complex zeroes?
Ans. Yes, a polynomial can have complex zeroes. Complex zeroes occur when the solutions to the polynomial equation involve imaginary numbers. These zeroes often occur in polynomials with degree greater than or equal to two. Complex zeroes are important in understanding the behavior and graph of a polynomial function.
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