Question

at which root does the graph of f(x) = (x + 4)^6(x + 7)^5 cross the x-axis?
-7
-4
4
7

Explanation:

Step1: Recall Root Behavior Rule

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

Step2: Analyze Each Factor

• For the factor \( (x + 4)^6=(x-(-4))^6 \), the exponent \( n = 6 \) (even). So the graph touches the x - axis at \( x=-4 \).
• For the factor \( (x + 7)^5=(x-(-7))^5 \), the exponent \( n = 5 \) (odd). So the graph crosses the x - axis at \( x=-7 \).

Answer:

\( \boldsymbol{-7} \) (the option corresponding to -7)