# Find all the zeros of the polynomial P(x) 2×4-3×3-5×2+9x-3

Let’s start with the concept of “Find all the zeros of the polynomial (2x4-3x3-5x2+9x-3), it is given that two of its zeros are √3  and -√3.

## Find all the zeros of the polynomial p(x)= (2x4-3x3-5x2+9x-3), it is given that two of its zeros are √3  and -√3.

Solution:  √3  and -√3 are zeros of polynomial P(x) = 2x4-3x3-5x2+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

x2 – 3 = 0

(x2 – 3) is factor of P(x) = 2x4-3x3-5x2+9x-3.

Now,

(x2 – 3) is completely divisible by P(x) = 2x4-3x3-5x2+9x-3.

When (2x4-3x3-5x2+9x-3) is divided by (x2-3) to get (2x2-3x+1) as a quotient and 0 as a remainder.

Factorise q(x) = (2x2-3x+1)

q(x) = 2x2-3x+1

0  = 2x2-2x-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) = 2x4-3x3-5x2+9x-3 are √3 , -√3 , 1 and ½

Some important identity of Polynomial

1. (a+b)2 = a2 + b2 + 2ab
2. (a-b)2 = a2 + b2 – 2ab
3. a2 – b2 = (a+b)(a-b)
4. (a+b+c)2 = a2+b2+c2+2ab+2bc+2ca
5. (a+b)3 = a3+b3+3a2b+3ab2
6. (a-b)3 = a3+b3-3a2b+3ab2
7. a3+b3 = (a+b)(a2+b2-2ab)
8. a3-b3 = (a+b)(a2+b2+2ab)

Polynomials

An expression of the form p(x) = a0+a1x+a2x2+……+anxn, where an 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) = 2x3 + 5x2 -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).

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