Understanding the Fundamental Theorem of Algebra

What does the Fundamental Theorem of Algebra state about the roots of a polynomial?

• A polynomial of degree n cannot have complex roots.

• A polynomial of degree n has at most n real roots.

• A polynomial of degree n has exactly n real roots.

• A polynomial of degree n has exactly n roots, including complex roots.

Can a 7th degree polynomial have 9 real roots?

• No, it must have exactly 7 complex roots.

• No, it can have at most 7 real roots.

• Yes, but only if it has complex roots.

• Yes, if it intersects the x-axis 9 times.

Is it possible for a polynomial to have 6 real roots and 1 non-real root?

• No, because a polynomial can only have real roots.

• Yes, but only for polynomials of degree greater than 7.

• No, because non-real roots come in conjugate pairs.

• Yes, if the polynomial is of 7th degree.

How many non-real complex roots can a 7th degree polynomial have?

• It can have 7 non-real complex roots.

• It can have an even number of non-real complex roots.

• It cannot have non-real complex roots.

• It can have any number of non-real complex roots.

Why can't a polynomial have an odd number of non-real complex roots?

• Because odd numbers of roots are reserved for real roots only.

• Because non-real complex roots are not possible in polynomials.

• Because the Fundamental Theorem of Algebra prohibits it.

• Because non-real complex roots always come in pairs.

What are the possible numbers of real roots for a 7th degree polynomial?

• 7, 5, 3, or 1

• 6, 4, 2, or 0

• 7, 6, 5, 4, 3, 2, 1, or 0

• Only 7 or 0

Why does a cubic polynomial always have at least one real root?

• Because the Fundamental Theorem of Algebra says so.

• Because cubic polynomials cannot have an even number of roots.

• Because it can be graphed to intersect the x-axis at least once.

• Because it cannot have complex roots.

Can a quadratic function have one real root and one non-real root?

• Yes, but only in special cases.

• No, because quadratic functions always have two real roots.

• No, because non-real roots come in conjugate pairs.

• Yes, if the discriminant is zero.

What does it mean for complex roots to come in conjugate pairs?

• One root is twice the value of the other.

• One root is the negative of the other.

• One root is the inverse of the other.

• One root is the complex conjugate of the other, differing only in the sign of the imaginary part.

Why is the Fundamental Theorem of Algebra important for understanding polynomials?

• It proves that polynomials can only have real roots.

• It guarantees that every polynomial equation has at least one solution.

• It shows how to graph polynomials accurately.

• It provides a method for finding all roots of any polynomial.