Question

a polynomial function has a root of $-4$ with multiplicity $4$, a root of $-1$ with multiplicity $3$, and a root of $5$ with multiplicity $6$. if the function has a positive leading coefficient and is of odd degree, which could be the graph of the function?

Explanation:

Step1: Determine the degree's parity and leading coefficient

The polynomial has roots with multiplicities 4, 3, and 6. The degree is the sum of multiplicities: \(4 + 3+ 6=13\), which is odd. The leading coefficient is positive. For a polynomial with a positive leading coefficient and odd degree, as \(x\to+\infty\), \(f(x)\to+\infty\), and as \(x\to-\infty\), \(f(x)\to-\infty\).

Step2: Analyze the roots' behavior (multiplicity)

• Root at \(x = - 4\) with multiplicity 4 (even): The graph touches the \(x\)-axis and turns around at \(x=-4\).
• Root at \(x=-1\) with multiplicity 3 (odd): The graph crosses the \(x\)-axis at \(x = - 1\).
• Root at \(x = 5\) with multiplicity 6 (even): The graph touches the \(x\)-axis and turns around at \(x = 5\).

Step3: Match with the graphs

• Check end - behaviors: We need a graph where as \(x\to+\infty\), \(y\to+\infty\) and as \(x\to-\infty\), \(y\to-\infty\).
• Check the roots' behavior: At \(x=-4\), the graph should touch and turn; at \(x = - 1\), cross; at \(x = 5\), touch and turn.

Looking at the four graphs:

• The first graph (top - left) has end - behavior as \(x\to-\infty\), \(y\to-\infty\) and \(x\to+\infty\), \(y\to+\infty\), touches at \(x=-4\) (since it just touches and turns), crosses at \(x=-1\) (or near - 1, considering the graph's crossing), and touches at \(x = 5\) (the graph touches the \(x\)-axis at \(x = 5\) and turns). The other graphs either have incorrect end - behaviors (e.g., the bottom - left has \(x\to+\infty\), \(y\to-\infty\) which is wrong for positive leading coefficient and odd degree) or incorrect root - crossing/touching behavior.

Answer:

The top - left graph (the first graph among the four)