Choose each statement that could describe all the zeros of f

Question

The polynomial f(x) has degree 4. Choose each statement that could describe all the zeros of f

1. One zero with multiplicity 3
2. One zero with multiplicity 4
3. One zero with multiplicity 5
4. One zero with multiplicity 1 and one zero with multiplicity 2
5. One zero with multiplicity 1 and one zero with multiplicity 3
6. One zero with multiplicity 2 and two zeros with multiplicity 1
7. One zero with multiplicity 2 and three zeros with multiplicity 1
8. Two zeros with multiplicity 2
9. Two zeros with multiplicity 2 and two zeros with multiplicity 1
10. Two zeros with multiplicity 2 and three zeros with multiplicity 1
11. Two zeros with multiplicity 1
12. Three zeros with multiplicity 1
13. Four zeros with multiplicity 1
14. Five zeros with multiplicity 1

Answer 1

Jordan:

Since f(x) is a polynomial of degree 4, f(x) has 4 root provided we consider multiplicities and complex roots.

So each root's multiplicities must add to 4.

One zero with multiplicity 3 doesn't work because there's a root missing. Remember we need exactly 4.

One zero with multiplicity 4 works!

One zero with multiplicity 5 doesn't work because there's an extra root. We need exactly 4.

One zero with multiplicity 1 and one zero with multiplicity 2 doesn't work because 1+2 =3 ≠ 4

One zero with multiplicity 1 and one zero with multiplicity 3 works! (1+3=4)

One zero with multiplicity 2 and two zeros with multiplicity 1 works! (2+1+1=4)

One zero with multiplicity 2 and three zeros with multiplicity 1 doesn't work because 2+1+1+1=5≠4

Try to figure out the others. Let me know if you need any help!

Answer 2

A fourth degree polynomial shall have 4 zeros. Any option in which the number of zeros adds to 4 is true.