Question

describe the behavior at each zero of f(x)=(x - 5)^3(x - 1)(x + 4)^2. the graph crosses the x - axis at - 4, 1, and 5. the graph touches the x - axis at - 4, 1, and 5. the graph touches the x - axis at 1 and 5 and crosses the x - axis at - 4. the graph touches the x - axis at - 4 and crosses the x - axis at 1 and 5.

Explanation:

Step1: Recall root - behavior rule

If \(f(x)=(x - a)^n\), when \(n\) is odd, the graph of \(y = f(x)\) crosses the \(x\) - axis at \(x=a\), and when \(n\) is even, the graph of \(y = f(x)\) touches the \(x\) - axis at \(x = a\).
For \(f(x)=(x - 5)^3(x - 1)(x + 4)^2\), the zeros are \(x=-4\), \(x = 1\), and \(x = 5\).

Step2: Analyze the zero \(x=-4\)

Since the factor corresponding to \(x=-4\) is \((x + 4)^2\) (where \(n = 2\) is even), the graph touches the \(x\) - axis at \(x=-4\).

Step3: Analyze the zero \(x = 1\)

Since the factor corresponding to \(x = 1\) is \((x - 1)^1\) (where \(n = 1\) is odd), the graph crosses the \(x\) - axis at \(x = 1\).

Step4: Analyze the zero \(x = 5\)

Since the factor corresponding to \(x = 5\) is \((x - 5)^3\) (where \(n = 3\) is odd), the graph crosses the \(x\) - axis at \(x = 5\).

Answer:

The graph touches the \(x\) - axis at \(-4\) and crosses the \(x\) - axis at \(1\) and \(5\).