Find all the zeroes of the polynomial 2x^4 3x^3 5x^2 + 9x 3, if two of its zeroes are √ 3 and √ 3

If we divide the equation (1) by the above quadratic by long division method we get another quadratic which is a factor of equation (1)
#:. (2x^4-3x^3-5x^2+9x-3)/(x^2-3)#, we get dividend as
#2x^2-3x+1#

To find factors of second quadratic we use split the middle term method
#2x^2-2x-x+1#, paring and taking out the common factors we get
#2x(x-1)-(x-1)#
#=>(x-1)(2x-1)#
Setting each factor #=0#, we obtain remaining two zeros as
#x=1, 1/2#

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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).

I hope you like this post Find all the zeros of the polynomial P(x) = 2x4-3x3-5x2+9x-3

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The given polynomial is f(x) = `2x^4 – 3x^3 – 5x^2 + 9x – 3`Since √3 and –√3 are the zeroes of f(x), it follows that each one of `(x – sqrt3) `and `(x + sqrt3)`is a factor of f(x).Consequently, `(x – sqrt3) (x + sqrt3)` = (x2 – 3) is a factor of f(x).

On dividing f(x) by (x2 – 3), we get:  

`f(x) = 0``⇒ 2x^4 – 3x^3 – 5x2 + 9x – 3 = 0``⇒ (x^2 – 3) (2x^2– 3x + 1) = 0``⇒ (x^2 – 3) (2x2– 2x – x + 1) = 0``⇒ (x – sqrt3) (x + sqrt3) (2x – 1) (x – 1) = 0``⇒ x = sqrt3 or x = -sqrt3 or x = 12 or x = 1`

Hence, all the zeroes are `sqrt3, -sqrt3`, 12 and 1.