Given:
The zeros of the polynomial are [tex]-2,\pm 5,4[/tex].
Degree = 4
Leading coefficient = 1
To find:
The polynomial.
Solution:
If c is a zero of a polynomial, then (x-c) must be a factor of the polynomial.
Here, -2,4,-5, 5, are zeros of the required polynomial, so (x+2), (x-4), (x+5), (x-5) are factors of required polynomial.
[tex]p(x)=(x+2)(x-4)(x+5)(x-5)[/tex]
[tex]p(x)=(x^2-4x+2x-8)(x^2-5^2)[/tex] [tex][\because a^2-b^2=(a-b)(a+b)][/tex]
[tex]p(x)=(x^2-2x-8)(x^2-25)[/tex]
Using distributive property, we get
[tex]p(x)=x^2(x^2-25)-2x(x^2-25)-8(x^2-25)[/tex]
[tex]p(x)=x^4-25x^2-2x^3+50x-8x^2+200[/tex]
On combining like terms, we get
[tex]p(x)=x^4-2x^3+(-25x^2-8x^2)+50x+200[/tex]
[tex]p(x)=x^4-2x^3-33x^2+50x+200[/tex]
Here, the leading coefficient is 1. So, it is the required polynomial.
Therefore, the correct option is E.
