Question

find the zeros for the following polynomial function and give the multiplicity for each zero. state whether the graph crosses the x - axis, or touches the x - axis and turns around, at each zero.
f(x)=2(x - 4)(x + 6)^3
the zero(s) of f is/are 4, - 6. (use a comma to separate answers as needed.)
the multiplicity of the smallest zero is 3. (type a whole number.)
the multiplicity of the largest zero is 1. (type a whole number.)
determine whether the graph crosses the x - axis or touches the x - axis and turns around, at the leftmost zero.
a. touches the x - axis and turns around
b. crosses the x - axis

Explanation:

Step1: Recall polynomial - zero rules

For a polynomial in factored form \(y = a(x - r_1)^{m_1}(x - r_2)^{m_2}\cdots(x - r_n)^{m_n}\), the zeros are \(r_i\) and the multiplicities are \(m_i\). If the multiplicity \(m\) of a zero \(r\) is odd, the graph of the polynomial crosses the \(x -\)axis at \(x = r\), and if \(m\) is even, the graph touches the \(x -\)axis and turns around at \(x = r\).

Step2: Identify the left - most zero and its multiplicity

The given polynomial is \(f(x)=2(x - 4)(x + 6)^{3}\). The zeros are \(x = 4\) and \(x=-6\). The left - most zero is \(x=-6\) and its multiplicity \(m = 3\) (since the exponent of \((x + 6)\) is 3).

Step3: Determine the graph behavior

Since the multiplicity of \(x=-6\) is 3 (an odd number), the graph of the function crosses the \(x -\)axis at \(x=-6\).

Answer:

B. crosses the x - axis