Question

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

Explanation:

Step1: Recall root crossing 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 at \( x = a \).

Step2: Analyze each factor

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

Answer:

-7