Question

write the equation of the graph shown below in factored form.
$f(x) = (x - 1)^2(x - 1)(x + 3)$
$f(x) = (x - 1)^2(x + 1)(x - 3)$
$f(x) = (x + 1)^2(x + 1)(x + 3)$
$f(x) = (x - 1)^2(x - 1)(x - 3)$

Explanation:

Step1: Identify x-intercepts

The graph touches or crosses the x - axis at \(x = - 1\), \(x = 1\) (with a touch, so even multiplicity), and \(x = 3\).

Step2: Analyze factors from x-intercepts

For a root \(x = a\), the factor is \((x - a)\). For \(x=-1\), the factor is \((x + 1)\); for \(x = 1\) (even multiplicity, let's assume multiplicity 2 for now), the factor is \((x - 1)^2\); for \(x = 3\), the factor is \((x - 3)\).
So the function in factored form should be \(f(x)=(x - 1)^2(x + 1)(x - 3)\) (we check the options, and this matches the second option).

Answer:

B. \(f(x)=(x - 1)^2(x + 1)(x - 3)\)