Suppose that a polynomial function of degree 5 with rational coefficients
has 0 (with multiplicity 2), 6, and –2 + 3i as zeros. Find the remaining zero.
A. –6
B. –2 – 3i
C. 0
D. 2 + 3i
B. complex roots of polynomials always occur as complex conjugate pairs
To find the remaining zero of the polynomial function, we can use the fact that complex roots always come in conjugate pairs.
Since the given polynomial has the complex root -2 + 3i, the conjugate of this root would also be a root. The conjugate of -2 + 3i is -2 - 3i.
Therefore, we have found three roots so far: 0 (with multiplicity 2), 6, -2 + 3i, -2 - 3i.
A polynomial of degree 5 will have 5 roots in total. Since we have accounted for 4 of the roots, there is one remaining root to find.
To find the remaining root, we can use the fact that the coefficients of the polynomial are rational. This means that the remaining root must also be rational.
The only remaining rational option from the given choices is A. -6.
So, the correct answer is A. -6.