Question

Construct a polynomial function with the following properties: third degree, only real coefficients, – 3 and2 + i are two of the zeros, y-intercept is – 75.

Answer 1

`-5x^3+5x^2+35x-75`

Explanation:

We can calculate the polynomial function like this:

`f(x)=a\cdot(x-x_1)\cdot(x-x_2)\cdot(x-x_3)`

We replace the zeros, to construct the polynomial, like this:

`\begin{gathered} f(x)=a\cdot(x-(-3))\cdot(x-(2+i))\cdot(x-(2-i)) \\ f(x)=a\cdot(x+3)\cdot(x-2-i)\cdot(x-2+i) \\ (x-2-i)\cdot(x-2-i)=x^2-2x+ix-2x+4-2i-ix+2i+i^2=x^2-4x+5 \\ f(x)=a\cdot(x+3)\cdot(x^2-4x+5) \\ (x+3)\cdot(x^2-4x+5)=x^3-4x^2+5x+3x^2-12x+15=x^3-x^2-7x+15 \\ f(x)=a\cdot(x^3-x^2-7x+15) \end{gathered}`

We plug in y-intercept to calculatea a:

`\begin{gathered} -75=a\cdot(0^3-0^2-7\cdot0+15)^{} \\ 15a=-75 \\ a=-(75)/(15) \\ a=-5 \\ \text{ replacing} \\ f(x)=-5\cdot\mleft(x^3-x^2-7x+15\mright) \\ f(x)=-5x^3+5x^2+35x-75 \end{gathered}`