Question

what are the possible outcomes when you apply the fundamental theorem of algebra to a polynomial equation of degree 4? (choose all that apply.) (2 points) four real roots and no complex roots three real roots and one complex root two real roots and two complex roots one real roots and three complex roots no real roots and four complex roots

Explanation:

Step1: Recall Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra states that a polynomial of degree \( n \) has exactly \( n \) roots (counting multiplicities) in the complex number system. For a degree 4 polynomial, there are 4 roots in total (real or complex, with complex roots occurring in conjugate pairs).

Step2: Analyze each option

• Four real roots and no complex roots: Valid, since all 4 roots can be real (complex roots would be zero, and real numbers are a subset of complex numbers).
• Three real roots and one complex root: Invalid, because complex roots must come in conjugate pairs (so the number of non - real complex roots must be even). Having 1 complex root violates this.
• Two real roots and two complex roots: Valid, the two complex roots form a conjugate pair, and the two real roots are also complex (since real numbers are complex) but are real. The total number of roots is \( 2 + 2=4 \).
• One real root and three complex roots: Invalid, because the number of non - real complex roots (3) is odd, but complex roots must come in conjugate pairs (so the number of non - real complex roots should be even).
• No real roots and four complex roots: Valid, the four complex roots can be two pairs of conjugate complex roots. The total number of roots is 4.

Answer:

A. Four real roots and no complex roots, C. Two real roots and two complex roots, E. No real roots and four complex roots