Answer:
[tex]P(x)=x^{4}-6x^{3}-16x^{2}+96x[/tex]
Step-by-step explanation:
1. The polynomial must have the following zeros:
[tex]x=-4\\x=0\\x=4\\x=6[/tex]
2. This means the following:
[tex]x+4=0\\x=0\\x-4=0\\x-6=0[/tex]
3. Multiply each term. The product of the multiplicaction is equal to zero. Then:
[tex](x+4)(x)(x-4)(x-6)=0[/tex]
4. [tex](x+4)[/tex] and [tex](x-4)[/tex] are conjugates, therefore, you have:
[tex](x^{2}-16)(x)(x-6)=0[/tex]
5. Apply the Distributive property. Then, you obtain:
[tex]x^{4}-6x^{3}-16x^{2}+96x=0[/tex]
6. The polynomial of degree 4 and zeros −4, 0, 4 and 6 is:
[tex]P(x)=x^{4}-6x^{3}-16x^{2}+96x[/tex]
