Question

recovery
graphing polynomial functions
at which root does the graph of $f(x) = (x - 5)^3(x + 2)^2$ touch the $x$-axis?
5 -2
-5 2

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's exponent

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

Answer:

\(-2\)