**Let’s start with the concept of “Find all the zeros of the polynomial (2x ^{4}-3x^{3}-5x^{2}+9x-3), it is given that two of its zeros are √3 and -√3.**

**Find all the zeros of the polynomial p(x)= (2x**^{4}-3x^{3}-5x^{2}+9x-3), it is given that two of its zeros are √3 ** and -√3****.**

^{4}-3x

^{3}-5x

^{2}+9x-3), it is given that two of its zeros are √3

Solution: √3 and -√3 are zeros of polynomial P(x) = 2x^{4}-3x^{3}-5x^{2}+9x-3.

x = √3 or x = -√3

x -√3 = 0 or x + √3 =0

(x -√3 )(x +√3 ) = 0 x 0

(x)^{2} – (√3 )^{2} = 0

x^{2 }– 3 = 0

(x^{2 }– 3) is factor of P(x) = 2x^{4}-3x^{3}-5x^{2}+9x-3.

Now,

(x^{2 }– 3) is completely divisible by P(x) = 2x^{4}-3x^{3}-5x^{2}+9x-3.

When (2x^{4}-3x^{3}-5x^{2}+9x-3) is divided by (x^{2}-3) to get (2x^{2}-3x+1) as a quotient and 0 as a remainder.

Factorise q(x) = (2x^{2}-3x+1)

q(x) = 2x^{2}-3x+1

0 = 2x^{2}-2x-x+1

0 = 2x(x-1) -1(x-1)

0 = (2x-1) (x-1)

Either,

2x-1 = 0 or x-1 = 0

X = ½ or x = 1

Hence, all zeros of polynomial P(x) = 2x^{4}-3x^{3}-5x^{2}+9x-3 are √3 , -√3 , 1 and ½

**Some important identity of Polynomial**

- (a+b)
^{2}= a^{2}+ b^{2}+ 2ab - (a-b)
^{2}= a^{2}+ b^{2}– 2ab - a
^{2}– b^{2}= (a+b)(a-b) - (a+b+c)
^{2}= a^{2}+b^{2}+c^{2}+2ab+2bc+2ca - (a+b)
^{3}= a^{3}+b^{3}+3a^{2}b+3ab^{2} - (a-b)
^{3}= a^{3}+b^{3}-3a^{2}b+3ab^{2} - a
^{3}+b^{3}= (a+b)(a^{2}+b^{2}-2ab) - a
^{3}-b^{3}= (a+b)(a^{2}+b^{2}+2ab)

** **

**Polynomials**

An expression of the form p(x) = a_{0}+a_{1}x+a_{2}x^{2}+……+a_{n}x^{n}, where a_{n} is not equal to zero, is called a polynomial in x of degree n.

^{ }

**Degree of polynomials**

If P(X) is a polynomial in x, the highest power of x in P(x) is called the degree of the polynomial P(x).

Ex: The degree of polynomial P(X) = 2x^{3} + 5x^{2} -7 is 3 because the degree of a polynomial is the highest power of polynomial.

**Zero of polynomial **

If the value of P(x) at x = K is zero then K is called a zero of the polynomial P(x).

**I hope you like this post Find all the zeros of the polynomial P(x) = 2x ^{4}-3x^{3}-5x^{2}+9x-3**