Question

Why is it important to consider multiplicity when determining the roots of a polynomial equation? Write your response, citing mathematical reasons, and providing examples that you create before reading the responses of your peers.

a. Multiplicity influences the behavior of the polynomial near its roots, affecting factors like tangency and bouncing off the x-axis.
b. Multiplicity has no impact on polynomial roots.
c. Multiplicity only matters for linear equations.
d. Multiplicity is crucial for real roots but not for complex roots.

Answer 1

Multiplicity is important to consider when determining the roots of a polynomial equation because it influences the behavior of the polynomial near its roots.

Step-by-step explanation:

Multiplicity is important to consider when determining the roots of a polynomial equation because it influences the behavior of the polynomial near its roots. The multiplicity of a root determines whether the polynomial crosses or touches the x-axis at that root. It affects factors like tangency and bouncing off the x-axis.

For example, if a polynomial has a root with multiplicity 1, it will cross the x-axis at that root. But if the root has multiplicity greater than 1, the polynomial will touch the x-axis at that root without crossing it.

Consider the polynomial equation (x - 2)(x + 3) = 0. It has two roots: x = 2 and x = -3. The root x = 2 has multiplicity 1 because it appears once in the factored form, while the root x = -3 has multiplicity 1 as well. Therefore, the polynomial crosses the x-axis at both roots.