Question

which equation choice could represent the graph shown below?
graph of a polynomial function with x-intercepts at -5 (crossing) and 2 (touching), and y-intercept positive
answer

• ( f(x) = (x + 5)^2(x - 2)^2 )
• ( f(x) = (x - 5)(x + 2)^2 )
• ( f(x) = (x - 5)^2(x + 2)^2 )
• ( f(x) = (x + 5)(x - 2)^2 )

Explanation:

Step1: Analyze x-intercepts

The graph crosses the x - axis at \(x=-5\) (with a single root, so odd multiplicity) and touches the x - axis at \(x = 2\) (with a double root, so even multiplicity).

Step2: Analyze the form of the function

For a root \(x = a\) with multiplicity \(m\), the factor is \((x - a)^m\). For \(x=-5\) (multiplicity 1), the factor is \((x + 5)\); for \(x = 2\) (multiplicity 2), the factor is \((x - 2)^2\). So the function should be of the form \(f(x)=(x + 5)(x - 2)^2\).

Step3: Eliminate other options

• Option \(f(x)=(x + 5)^2(x - 2)^2\): Both roots have even multiplicity, but the graph crosses at \(x=-5\) (should have odd multiplicity), so eliminate.
• Option \(f(x)=(x - 5)(x + 2)^2\): The roots are \(x = 5\) and \(x=-2\), which do not match the x - intercepts of the graph, eliminate.
• Option \(f(x)=(x - 5)^2(x + 2)^2\): Roots are \(x = 5\) and \(x=-2\), do not match the graph's x - intercepts, eliminate.

Answer:

\(f(x)=(x + 5)(x - 2)^2\) (the last option: \(f(x)=(x + 5)(x - 2)^2\))