Given:
Degree of a polynomial = 3
Zeros of the polynomial = 0,5,8
f(10)=17
To find:
The polynomial function.
Solution:
The general form of a polynomial is
[tex]P(x)=a(x-c_1)^{m_1}(x-c_2)^{m_2}...(x-c_n)^{m_n}[/tex]
Where, a is a constant, [tex]c_1,c_2,...,c_n[/tex] are zeros with multiplicity [tex]m_1,m_2,...,m_n[/tex] respectively.
Since, 0,5,8 are zeros of the polynomial, therefore, (x-0),(x-5), (x-8) are the factors of required polynomial.
[tex]f(x)=a(x)(x-5)(x-8)[/tex] ...(i)
Putting x=10, we get
[tex]f(10)=a(10)(10-5)(10-8)[/tex]
We have f(10)=17.
[tex]17=a(10)(5)(2)[/tex]
[tex]17=a(100)[/tex]
[tex]\dfrac{17}{100}=a[/tex]
[tex]0.17=a[/tex]
Putting a=0.17 in (i).
[tex]f(x)=0.17(x)(x-5)(x-8)[/tex]
Therefore, the required polynomial is [tex]f(x)=0.17(x)(x-5)(x-8)[/tex].
