Answer:
[tex]x^{3} +3x^{2}-9x+5 = 0[/tex]
Step-by-step explanation:
We are given that the only roots of the given polynomial f(x) are 1 and -5 with multiplicity (the number of times the roots are repeating) 2 and 1 respectively.
Also, it is provided that the function satisfy that f(x)=0.
So, comparing the two information we have that,
f(x) = 0 and 1, -5 are the roots of f(x),
i.e [tex](x-1)^{2} \times (x+5) = 0[/tex]
i.e. [tex](x^{2} +1-2x) \times (x+5) = 0[/tex]
i.e. [tex]x^{3} +3x^{2}-9x+5 = 0[/tex]
Also, we can see that ths polynomial has leading co-efficient 1 i.e. the co-efficient of highest degree variable i.e. x^{3}.
Hence, the polynomial having leading co-efficient 1 and roots 1, -5 is [tex]x^{3} +3x^{2}-9x+5 = 0[/tex].
