# Zeros of polynomial

For a polynomial, there could be some values of the variable for which the polynomial will be zero. These values are called **zeros of polynomial**. General form of a polynomial in x is a_{n}x^{n} + a_{n-1}x^{n-1} +….. + a_{1}x + a_{0}, where a_{n}, a_{n-1}, ….. , a_{1}, a_{0} are constants, a_{n }≠0 and n is a whole number. For example, algebraic expressions such as √x + x + 5, x^{2} + 1/x** ^{2}** are not polynomials because all exponents of x in terms of the expressions are not whole numbers.

## Finding Zeros of Polynomials

Zeros of a polynomial can be defined as the points where the polynomial becomes zero on the whole. A polynomial having value zero (0) is called zero polynomial. Degree of a polynomial is the highest power of the variable x.

- Polynomial of degree 1 is known as linear polynomial.

Standard form is ax + b, where a and b are real numbers and a≠0.

2x + 3 is a linear polynomial. - Polynomial of degree 2 is known as quadratic polynomial.

Standard form is ax^{2}+ bx + c, where a, b and c are real numbers and a ≠ 0

x^{2}+ 3x + 4 is an example for quadratic polynomial. - Polynomial of degree 3 is known as a cubic polynomial.

Standard form is ax^{3}+ bxx^{2}+ cx + d, where a, b, c and d are real numbers and a≠0.

x^{3}+ 4x + 2 is an example for cubic polynomial.

Similarly,

y^{6} + 3y^{4} + y is a polynomial in y of degree 6.

### Examples

**Example: What is the value of a if degree of polynomial x ^{3} + x^{a-4} + x^{2} + 1 is 4?**

Degree of a polynomial P(x) is the highest power of x in P(x).

Therefore, **x ^{a-4} ** =

**x**

^{4}a-4 = 4, a = 4+4 =8

Consider, P(x)= x^{2} – 3x + 2,

Put x = 3 in P(x) which gives,

P(3) = 9 – 9 + 2 = 2

Replace x by 2 in the polynomial x^{2} – 3x + 2, which gives P(2) = 4 – 6 + 2 = 0

Similarly, value of x^{2} – 3x + 2, at x = 0 is,

P(0) = 0-0+2 = 2

In general; if P(x) is a polynomial in x and k is any real number, then value of P(k) at x = k is denoted by P(k) is found by replacing x by k in P(x).

In the polynomial x^{2} – 3x + 2,

Replacing x by 1 gives,

P(1) = 1 – 3 + 2 = 0

Similarly, replacing x by 2 gives,

P(2) = 4-6+2 = 0

For a polynomial P(x), real number k is said to be zero of polynomial P(x), if P(k) = 0.

Therefore, 1 and 2 are the zeros of polynomial x^{2} – 3x + 2.

## Zeros of Polynomial Formula

Consider, P(x) = 4x + 5 to be a linear polynomial in one variable.

Let a be zero of P(x), then,

P(a) = 4k+5 = 0

Therefore, k = -5/4

In general, If k is zero of the linear polynomial in one variable; P(x) = ax +b, then

P(k) = ak+b = 0

k = -b/a

It can also be written as,