Question

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. Write the polynomial in standard form. 2,5-i

Answer 1

Answer:

Hence, the polynomial function in standard form is:

$f(x)=x^3 - 12x^2 + 46x - 52$

Step-by-step explanation:

We are given a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros 2,5-i.

As we know that the polynomial function has rational coefficients.

so any complex root always appear in pair.

so as 5-i is a root of f(x) then it's complex conjugate 5+i is also an root of f(x).

Hence f(x) could be represented as:  

$f(x)= (x-2)[x-(5-i)][x-(5+i)]\\\\f(x) = (x - 2)[(x-5)+ i][(x-5)- i)\\\\f(x)= (x - 2)[(x-5)^2 - i^2]\\\\f(x)=(x - 2)(x^2 - 10x + 25 + 1)\\\\f(x)= (x - 2)(x^2 - 10x + 26)\\\\f(x) = x^3 - 10x^2 + 26x - 2x^2 + 20x-52\\\\f(x)=x^3 - 12x^2 + 46x - 52$

Hence, the polynomial function in standard form is:

$f(x)=x^3 - 12x^2 + 46x - 52$