Question

at which root does the graph of $f(x) = (x - 5)^3(x + 2)^2$ touch the x-axis?
$\circ$ $-5$
$\circ$ $-2$
$\circ$ $2$
$\circ$ $5$

Explanation:

Step1: Recall Root Behavior Rule

For a polynomial \( f(x) = (x - a)^n \), if \( n \) is even, the graph touches the x - axis at \( x = a \); if \( n \) is odd, the graph crosses the x - axis at \( x = a \).

Step2: Analyze Each Factor

• For the factor \( (x - 5)^3 \), the exponent \( n = 3 \) (odd), so the graph crosses the x - axis at \( x = 5 \).
• For the factor \( (x + 2)^2=(x-(-2))^2 \), the exponent \( n = 2 \) (even), so the graph touches the x - axis at \( x=-2 \).

Answer:

B. \(-2\) (assuming the option for \(-2\) is labeled as B, following the order of options: \(-5\), \(-2\), \(2\), \(5\))