Question

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

• 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:

-2