Question

Hello, May I please get some assistance with this homework question? I posted an image below Q7

Answer 1

A polynomial whose zeros are four has a degree of 4. Thus;

`\begin{gathered} x=-3 \\ x+3=0\ldots(1) \\ x=-2 \\ x+2=0\ldots\text{.}(2) \\ x=1 \\ x-1=0\ldots\text{.}(3) \\ x=3 \\ x-3=0\ldots(4) \end{gathered}`

Now, we would multiply the factors;

`\begin{gathered} (x+3)(x+2)(x-1)(x-3) \\ \end{gathered}`

We have;

`\begin{gathered} (x+3)(x+2)=x\mleft(x+2\mright)+3\mleft(x+2\mright) \\ (x+3)(x+2)=x^2+2x+3x+6 \\ (x+3)(x+2)=x^2+5x+6 \end{gathered}`
`\begin{gathered} (x+3)(x+2)(x-1)=(x^2+5x+6)(x-1) \\ (x+3)(x+2)(x-1)=x^3+4x^2+x-6 \end{gathered}`
`\begin{gathered} (x+3)(x+2)(x-1)(x-3)=(x^3+4x^2+x-6)(x-3) \\ (x+3)(x+2)(x-1)(x-3)=x^4+x^3-11x^2-9x+18 \end{gathered}`

FINAL ANSWER:

`f(x)=x^4+x^3-11x^2-9x+18`