Answer:
[tex]\large\boxed{p(x)=x^6-7x^5+69x^4/4-35x^3/2+25x^2/4}[/tex]
Explanation:
The zeros of the graph of a polynomial function, [tex]p(x),[/tex] are the roots of the polynomial or, in other words, the solutions to the equation [tex]p(x)=0[/tex] .
Given the zeros of a polynomial function, you can write the polynomial in a factored form:
[tex]p(x)=A(x-x_1)(x-x_2)(x-x_1)...(x-x_n)[/tex]
Where [tex]x_1,x_2,x_3,...,x_n[/tex] are the zeroes or roots of the polynomial.
The muliplicity of a zero is the number of times that factor with that root appears in the polynomial or, what is the same, the exponent of such factor.
Thus, for the zeros 0 (multiplicity 2), 1 (multiplicity 2), and 5/2 (multiplicity 2), you can write the polynomial as:
[tex]p(x)=A(x-0)^2(x-1)^2(x-5/2)^2[/tex]
The standard form of a polynomial is [tex]ax^n+bx^{n-1}+...+mx+n[/tex]
Hence, to write our polynomial function in standard form, use any value of A (I will use A = 1) and expand the polynomial:
[tex]p(x)=x^2(x-1)^2(x-5/2)^2[/tex]
[tex]p(x)=x^2(x^2-2x+1)(x^2-5x+25/4)[/tex]
[tex]p(x)=(x^4-2x^3+x^2)(x^2-5x+25/4)[/tex]
[tex]p(x)=x^6-5x^5+25x^4/4-2x^5+10x^4-25x^3/2+x^4-5x^3+25x^2/4[/tex]
Now, you just must add the like terms:
[tex]p(x)=x^6-7x^5+69x^4/4-35x^3/2+25x^2/4[/tex]
