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: Calculate polynomial degree

Degree = $4 + 3 + 1 = 8$
Wait, correction: The problem states odd degree, so the root at 5 must have odd multiplicity (we assume multiplicity 1 here, making total degree $4+3+1=8$ is even, so adjust: root 5 has multiplicity 3, total degree $4+3+3=10$ no, wait: odd degree means sum of multiplicities is odd. 4 (even) +3 (odd) =7, so root 5 needs even multiplicity? No, 4+3+2=9 (odd). Wait, key:

Step1: Identify end behavior

Positive leading coefficient, odd degree: as $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to-\infty$.

Step2: Analyze root behavior

• Root $x=-4$, multiplicity 4 (even): graph touches x-axis and turns around here.
• Root $x=-1$, multiplicity 3 (odd): graph crosses x-axis here, with a "flattened" shape (higher odd multiplicity).
• Root $x=5$, multiplicity (odd, to make total degree odd): graph crosses x-axis here.

Step3: Match to graphs

Top graph:

• As $x\to-\infty$, $f(x)\to+\infty$ (wrong for odd positive leading coefficient).

Bottom graph:

• As $x\to+\infty$, $f(x)\to+\infty$; as $x\to-\infty$, $f(x)\to-\infty$ (correct end behavior).
• Has a root where graph touches (even multiplicity), a root where it crosses with flattening (odd multiplicity), and a root where it crosses (odd multiplicity), matching the given roots.

Answer:

The bottom graph (the lower of the two provided graphs, with end behavior falling to the left and rising sharply to the right, showing a touch at an x-intercept, a flattened crossing, and a sharp crossing) is the correct graph.