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Question 6 (1 point)
state the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots.
a # of complex roots: 6
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible rational roots:
Roots:
b # of complex roots: 6
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible rational roots:
Roots:
c # of complex roots: 3
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible rational roots:
Roots
d # of complex roots: 3
Possible # of real roots: 3 or 1
Possible # of imaginary roots: 2 or 0
Possible rational roots:
Roots:

Ask by Zimmerman Crawford. in the United States
Mar 17,2025

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Tutor-Verified Answer

Answer

The polynomial has 3 complex roots. It can have 3 real roots or 1 real root and 2 imaginary roots. The possible rational roots are . The roots are , , and .

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Mind Expander

When diving into the world of polynomial equations, it’s essential to note that a cubic equation like will always have three roots, some of which could be real or complex. The Fundamental Theorem of Algebra tells us that the number of roots (counting multiplicities) equals the degree of the polynomial, which in this case is 3. So, buckle up – the possibilities are thrilling and diverse!
When looking for roots, the Rational Root Theorem is your new best friend! It gives you a yummy list of potential rational roots to test: . Simply plug these values into the polynomial, and if it equals zero, you’ve struck root gold! But remember, not all polynomials are as straightforward as they seem – keep an eye out for those pesky irrational or complex ones!

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