Question

graph the polynomial function f(x)=x^3 + x^2 - 6x. answer parts (a) through (e). crosses or touches the x - axis at each x - intercept. the zero(s) of f is/are 0, - 3,2. (type an integer or a simplified fraction. use a comma to separate answers as needed. type each answer only once.) the smallest zero is of multiplicity, so the graph of f the x - axis at x =. the middle zero is of multiplicity, so the graph of f the x - axis at x =. the largest zero is of multiplicity, so the graph of f the x - axis at x =.

Explanation:

Step1: Factor the polynomial

First, factor \(f(x)=x^{3}+x^{2}-6x=x(x^{2}+x - 6)=x(x + 3)(x - 2)\). The zeros are found by setting \(f(x)=0\), so \(x=0\), \(x=-3\), \(x = 2\). The smallest zero is \(x=-3\), the middle zero is \(x = 0\), and the largest zero is \(x=2\).

Step2: Determine multiplicity

Since the factors \(x\), \(x + 3\), and \(x - 2\) all have an exponent of 1 (implicitly), each zero has a multiplicity of 1.

Step3: Analyze behavior at x - intercepts

If the multiplicity of a zero \(a\) of a polynomial function \(y = f(x)\) is odd, the graph of the function crosses the \(x\) - axis at \(x=a\). Since the multiplicity of each of \(x=-3\), \(x = 0\), and \(x=2\) is 1 (an odd number), the graph of \(f(x)\) crosses the \(x\) - axis at \(x=-3\), \(x = 0\), and \(x=2\).

Answer:

The smallest zero \(x=-3\) is of multiplicity 1, so the graph of \(f\) crosses the \(x\) - axis at \(x=-3\). The middle zero \(x = 0\) is of multiplicity 1, so the graph of \(f\) crosses the \(x\) - axis at \(x = 0\). The largest zero \(x=2\) is of multiplicity 1, so the graph of \(f\) crosses the \(x\) - axis at \(x=2\).