Question

Consider the graph of m(x). Given that (x+1) is a factor of m(x), which of the following could be the multiplicity of (x+1) ? Option #1: −1 Option #2: 0 Option #3: 1 Option #4: 2

Answer 1

The multiplicity of a factor in a polynomial indicates how many times that factor appears in the polynomial. For the factor \((x + 1)\) to be part of \(m(x)\):

1. A multiplicity of **-1** is not possible, as multiplicity must be a non-negative integer.
2. A multiplicity of **0** would mean that \((x + 1)\) is not a factor of \(m(x)\) at all, which contradicts the given condition that \((x + 1)\) is a factor.
3. A multiplicity of **1** implies that \((x + 1)\) is a factor that appears once in the polynomial, meaning that \(m(-1) = 0\) and the polynomial touches the x-axis at this point.
4. A multiplicity of **2** means that \((x + 1)\) is a factor that appears twice in the polynomial, also leading to \(m(-1) = 0\) but indicating that the polynomial touches the x-axis and turns around at this point.

Thus, the possible multiplicities of \((x + 1)\) given that it is a factor of \(m(x)\) are **1 or 2**.

Therefore, the valid options for the multiplicity of \((x + 1)\) are:

- Option #3: **1**
- Option #4: **2**

The other options, -1 and 0, are not valid.