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:

Answer:

The degree of the polynomial is \(4 + 3+6=13\) (odd). For a polynomial \(y = a(x + 4)^{4}(x + 1)^{3}(x - 5)^{6}\) with \(a>0\). As \(x\to-\infty\), \(y=a(-\infty)^{4}(-\infty)^{3}(-\infty)^{6}=a(-\infty)\) (since \(4 + 3+6 = 13\) and \(a>0\)), so \(y\to-\infty\) as \(x\to-\infty\). As \(x\to+\infty\), \(y=a(+\infty)^{4}(+\infty)^{3}(+\infty)^{6}=a(+\infty)\), so \(y\to+\infty\) as \(x\to+\infty\). The root \(x=-4\) (multiplicity \(4\), even - graph touches the \(x\) - axis), \(x = - 1\) (multiplicity \(3\), odd - graph crosses the \(x\) - axis), \(x = 5\) (multiplicity \(6\), even - graph touches the \(x\) - axis). The second graph (from the left) is the correct one.