Question

a polynomial function has a root of −6 with multiplicity 3 and a root of 2 with multiplicity 4. if the function has a negative leading coefficient and is of odd degree, which could be the graph of the function?

Explanation:

Step1: Analyze the roots and multiplicities

The polynomial has a root \(x = - 6\) with multiplicity \(3\) (odd multiplicity) and a root \(x=2\) with multiplicity \(4\) (even multiplicity). For a polynomial, at a root with odd multiplicity, the graph crosses the \(x -\)axis, and at a root with even multiplicity, the graph touches the \(x -\)axis and turns around. So, the graph should cross the \(x -\)axis at \(x=-6\) and touch the \(x -\)axis at \(x = 2\).

Step2: Analyze the end - behavior

The degree of the polynomial is the sum of the multiplicities, \(3 + 4=7\) (odd degree). The leading coefficient is negative. For a polynomial of odd degree with a negative leading coefficient, as \(x
ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\).

Step3: Match with the graphs

• The top - left graph: Crosses the \(x -\)axis at \(x=-6\), touches the \(x -\)axis at \(x = 2\), and has the end - behavior: as \(x

ightarrow+\infty\), \(y
ightarrow-\infty\) and as \(x
ightarrow-\infty\), \(y
ightarrow+\infty\).

• The top - middle graph: Does not have a root at \(x=-6\) (it does not cross the \(x -\)axis at \(x = - 6\)) and does not touch the \(x -\)axis at \(x = 2\) (it crosses the \(x -\)axis at \(x=2\)), so it is incorrect.
• The top - right graph: Has a positive leading coefficient end - behavior (as \(x

ightarrow+\infty\), \(y
ightarrow+\infty\)) which is wrong for a negative leading coefficient, so it is incorrect.

• The bottom - left graph: Has a positive leading coefficient end - behavior (as \(x

ightarrow+\infty\), \(y
ightarrow+\infty\)) which is wrong for a negative leading coefficient, so it is incorrect.

Answer:

The graph on the Top - Left (the first graph among the four)