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Polynomials

    Zeroes of a Polynomial

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9th class > Polynomials > Zeroes of a Polynomial

Zeroes of a Polynomial

Consider the polynomial p(x) = 5x3-2x2 + 3x-2

Instructions

5x3−2x2+3x−2

  • If we replace x by 1 everywhere in p(x), we get
  • calculate the terms
  • Therfore, p(1) =
  • Also find the p(0) =
  • calculate the terms
  • Therfore, p(0) =

Example 2

Find the value of each of the following polynomials at the indicated value of variables:

(i) p(x) = 5x2-3x+7 at x = 1

Instructions

5x2−3x+7

  • The value of the polynomial p(x) at x = 1 is given by
  • calculate the terms
  • Therfore, p(x) =
  • We have found the answer.

(ii) q(y) = 3y3−4y+11 at y = 2

Instructions

3y3−4y+11

  • The value of the polynomial q(y) at y = 2 is given by
  • calculate the terms
  • Therfore, q(y) = +
  • We have found the answer.

(iii) p(t) = 4t4+5t3−t2+6 at t = a

Instructions

4t4+5t3−t2+6

  • The value of the polynomial p(t) at t = a is given by
  • Calculating the terms, we get: p(a) = + a3 + + 6.
  • We have found the answer.

Now, consider the polynomial p(x) = x – 1.

What is p(1)? Note that : p(1) = 1 – 1 = .

As p(1) = 0, we say that 1 is a zero of the polynomial p(x).

Similarly, you can check that 2 is a zero of q(x), where q(x) = x – 2.

In general, we say that a zero of a polynomial p(x) is a number c such that p (c) = 0. In other words,

Zeros of a Polynomial are those values when put in the polynomial instead of a variable, the result becomes zero.

You must have observed that the zero of the polynomial x – 1 is obtained by equating it to 0, i.e., x – 1 = 0, which gives x = .

We say p(x) = 0 is a polynomial equation and 1 is the root of the polynomial equation p(x) = 0. So we say 1 is the zero of the polynomial x – 1, or a root of the polynomial equation x – 1 = 0.

Now, consider the constant polynomial 5. Can you tell what its zero is?

It has no zero of polynomial because replacing x by any number in 5x0 still gives us 5.

In fact, a non-zero constant polynomial has no zero. What about the zeroes of the zero polynomial? By convention, every real number is a zero of the zero polynomial.

Note: Do not get confused between zeroes of polynomials and zero of polynomial

Example 3

Check whether –2 and 2 are zeroes of the polynomial x + 2

Instructions

Let p(x) = x + 2.
Then p(2) = 2 + 2 = , p(–2) = –2 + 2 =
Therefore, –2 is a zero of the polynomial x + 2, but 2 is not.

Example 4

Find a zero of the polynomial p(x) = 2x + 1.

Instructions

Finding a zero of p(x), is the same as solving the equation p(x) = 0
Now, 2x + 1 = 0 gives us x =
So, −12 is a zero of the polynomial 2x + 1.
Now, if p(x) = ax + b, a ≠ 0, is a linear polynomial, how can we find a zero of p(x)? Example 4 may have given you some idea. Finding a zero of the polynomial p(x), amounts to solving the polynomial equation p(x) = 0.
Now, p(x) = 0 means ax + b = 0, a ≠ 0
So, ax = –b i.e., x =
So, x = −ba is the only zero of p(x), i.e., a linear polynomial has one and only one zero.

Now: We can say that 1 is the zero of x – 1, and –2 is the zero of x + 2.

Example 5

Verify whether 2 and 0 are zeroes of the polynomial x2-2x

Instructions

Let p(x) = x2– 2x
Then p(2) = 22-4 = 4 - 4 =
and p(0) = 0 – 0 =
Hence, 2 and 0 are both zeroes of the polynomial x2– 2x

Let us now list our observations:

(i) A zero of a polynomial need not be 0.

(ii) 0 may be a zero of a polynomial.

(iii) Every linear polynomial has one and only one zero.

(iv) A polynomial can have more than one zero.