Final answer:
A 7th degree polynomial can have a combination of real and complex roots, including repeated roots. This can include 7 real roots, or a combination of real and complex roots that total to 7.
Step-by-step explanation:
A 7th degree polynomial can have a combination of real and complex roots, including repeated roots. For instance, a polynomial of degree 7 can have 7 real roots, or it can have a combination of real and complex roots that total to 7. This is due to the Fundamental Theorem of Algebra, which states that a n-degree polynomial produces n roots, either real or complex.
Examples include:
- 7 real roots: x, x, x, x, x, x, x
- 5 real and 1 complex pair: x, x, x, x, x, a + bi, a - bi
- 3 real and 2 complex pairs: x, x, x, a + bi, a - bi, c + di, c - di
Each pair of complex numbers (a + bi, a - bi) or (c + di, c - di) counts as two roots. Polyvalent roots are counted according to their multiplicity.
Learn more about Polynomial Roots