(a)

Question

Describe the combination of real and non-real complex/imaginary roots for each graphed function. Include the multiplicity of each real root.

Part A
Quartic - 4th degree

Answer 1

To describe the combination of real and non-real complex (imaginary) roots for a quartic (4th degree) polynomial, we first need to establish the general properties of such functions.

Key Concepts:

1. Roots of Polynomial Functions: A quartic polynomial can have up to 4 roots, which can be real (on the x-axis) or complex (not intersecting the x-axis).

2. Multiplicity of Roots: The multiplicity of a root refers to the number of times it is repeated as a root of the polynomial. For example, if a root \( r \) appears twice, it has a multiplicity of 2.

3. Complex Conjugate Root Theorem: If a polynomial has real coefficients and a complex root, then its conjugate is also a root. This implies that complex roots always come in pairs.

Describing Roots for a Quartic Function:

For a quartic polynomial, the possible combinations of real and complex roots, as well as their multiplicities, are as follows:

1. Four Real Roots:

   • Example: \( (x - r_1)(x - r_2)(x - r_3)(x - r_4) \)
   • Each \( r_i \) has multiplicity 1, or some may have higher multiplicities.
   • Total: 4 real roots.

2. Two Real Roots and Two Complex Roots:

   • Example: \( (x - r_1)(x - r_2)((x - a) + bi)((x - a) - bi) \)
   • Say \( r_1 \) has multiplicity 2: \( (x - r_1)^2(x - r_2)((x - a) + bi)((x - a) - bi) \)
   • Total: 2 real roots (with multiplicity) and 2 complex (non-real) roots.

3. One Real Root and Three Complex Roots:

   • Example: \( (x - r)((x - a) + bi)((x - a) - bi)(x - c) \)
   • One real root \( r \) can have any multiplicity (e.g., multiplicity 1), while three complex roots imply one double root.
   • Total: 1 real root (potentially with multiplicity) and 3 complex roots (counting complex conjugate pairs).

4. Four Complex Roots:

   • Example: \( ((x - a) + bi)((x - a) - bi)((x - c) + di)((x - c) - di) \)
   • In this case, all roots are non-real complex numbers.
   • Total: 0 real roots and 4 complex roots.

Conclusion:

To analyze a specific quartic function given values or a graph, you could apply the above combinations to determine the number of real and complex roots. Note the multiplicity of each real root, should there be any. If you have a particular function or graph to analyze, please provide the details for a more specific breakdown of its roots!