Question

section 6 – multiple choice
for each problem, circle the correct answer and show all the work it.
4 points each
15.
which of the following could be the equation of the
graph above?
a) $y = x(x - 2)(x + 3)$
b) $y = x^2(x - 2)(x + 3)$
c) $y = x(x + 2)(x - 3)$
d) $y = x^2(x + 2)(x - 3)$

Explanation:

Step1: Identify x-intercepts

The graph intersects the x - axis at \(x=-3\), \(x = 0\) (with a touch - and - turn behavior, indicating a double root), and \(x = 2\).

Step2: Analyze the roots

For a polynomial, if \(r\) is a root, then \((x - r)\) is a factor. At \(x = 0\), since the graph touches the x - axis and turns around, the multiplicity of the root \(x = 0\) is 2, so the factor is \(x^{2}\). For the root \(x=-3\), the factor is \((x + 3)\), and for the root \(x = 2\), the factor is \((x-2)\).
So the polynomial should be of the form \(y=x^{2}(x + 3)(x - 2)\) (which can be rewritten as \(y=x^{2}(x-2)(x + 3)\) by the commutative property of multiplication of factors).
Let's check the options:

• Option A: \(y=x(x - 2)(x + 3)\) has a single root at \(x = 0\) (multiplicity 1), but the graph has a double root at \(x = 0\), so A is incorrect.
• Option B: \(y=x^{2}(x - 2)(x + 3)\) has roots at \(x = 0\) (multiplicity 2), \(x=2\) and \(x=-3\), which matches the x - intercepts of the graph.
• Option C: \(y=x(x + 2)(x - 3)\) has roots at \(x = 0\) (multiplicity 1), \(x=-2\) and \(x = 3\), which do not match the x - intercepts of the graph, so C is incorrect.
• Option D: \(y=x^{2}(x + 2)(x - 3)\) has roots at \(x = 0\) (multiplicity 2), \(x=-2\) and \(x = 3\), which do not match the x - intercepts of the graph, so D is incorrect.

Answer:

B) \(y = x^{2}(x - 2)(x + 3)\)