**Zeros of a polynomial :**

Here we are going to see how to find zero of a polynomials.

If the value of a polynomial is zero for some value of the variable then that value is known as zero of the polynomial.

**Definition :**

Let p(x) be a polynomial in x. If p(x) = 0, then we say that a is a zero of the polynomial p(x).

Let us consider the example given below.

**Example 1 :**

Find the zeros of the following linear polynomial

p(x) = 2x + 3

**Solution :**

p(x) = 2x + 3

Now we have to think about the value of x, for which the given function will become zero.

For that let us factor out 2

p(x) = 2 (x + 3/2)

Instead of "x" , if we apply -3/2 p(x) will become zero.

Hence -3/2 is the zero of the given linear polynomial.

**Example 2 :**

Find the zeros of the following linear polynomial

p(x) = 4x - 1

**Solution :**

p(x) = 4x - 1

Now we have to think about the value of x, for which the given function will become zero.

For that let us factor out 4

p(x) = 4 (x - 1/4)

By applying the value 1/4 instead of x, the function p(x) will become zero.

Hence 1/4 is the zero of the given linear polynomial.

For a quadratic equation, there will be two zeros. In order to find those zeros, we may use the method called factoring.

**Example 3 :**

Find the zeros of the quadratic equation x² + 17 x + 60 by factoring.

**Solution :**

p(x) = x² + 17 x + 60

p(x) = x² + 12x + 5x + 60

p(x) = x (x + 12) + 5 (x + 12)

p(x) = (x + 5) (x + 12)

If x = -5

p(x) = (-5 + 5) (-5 + 12) = 0

If x = -12

p(x) = (-12 + 5) (-12 + 12) = 0

Hence the zeros are -5 and -12.

For a cubic equation, there will be three zeros. In order to find those zeros, we may use the methods

(i) Factor theorem

(ii) Synthetic division

**Example 4 :**

Find the zeros of the following polynomial 4 x³ - 7 x + 3

**Solution :**

Let p (x) = 4 x³ - 7 x + 3

x = 1

p (1) = 4 (1)³ -7 (1) + 3

= 4 - 7 + 3

= 7 - 7

= 0

So we can decide (x - 1) is a factor. To find other two factors, we have to use synthetic division.

So, the factors are (x - 1) and (4 x² - 4 x - 3). By factoring this quadratic equation we get (2 x + 3) (2 x - 1)

Hence the required three factors are (x - 1) (2 x + 3) (2 x - 1)

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