Question

Write a polynomial function in factored form with double roots at -2 and 5 and the constant term of the polynomial in standard form is -40.

Answer 1

`f(x)=-0.4x^4+2.4x^3+8.4x^2-24x-40`

Explanation:

If -2 and 5 are double roots of the polynomial function, then the function expression is

`f(x)=a(x-(-2))^2(x-5)^2\\ \\f(x)=a(x+2)^2(x-5)^2`

Rewrite this function in standard form:

`f(x)=a(x^2+4x+4)(x^2-10x+25)\\ \\f(x)=a(x^4-10x^3+15x^2+4x^3-40x^2+100x+4x^2-40x+100)\\ \\f(x)=a(x^4-6x^3-21x^2+60x+100)\\ \\f(x)=ax^4-6ax^3-21ax^2+60ax+100a`

The constant term of the polynomial in standard form is -40, so

`100a=-40\\ \\a=-(40)/(100)=-0.4`

Therefore, the function expression is

`f(x)=-0.4x^4-6\cdot (-0.4)x^3-21\cdot (-0.4)x^2+60\cdot (-0.4)x+100\cdot (-0.4)\\ \\f(x)=-0.4x^4+2.4x^3+8.4x^2-24x-40`