Question

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

Explanation:

Step1: Recall root and multiplicity rule

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

Step2: Find roots of \( f(x) = (x + 4)^6(x + 7)^5 \)

Rewrite the factors: \( (x + 4)^6=(x - (-4))^6 \) and \( (x + 7)^5=(x - (-7))^5 \). So the roots are \( x=-4 \) (multiplicity 6, even) and \( x = -7 \) (multiplicity 5, odd).

Step3: Determine crossing root

Since the multiplicity of \( x=-7 \) is odd (5), the graph crosses the \( x \)-axis at \( x=-7 \). The multiplicity of \( x=-4 \) is even (6), so the graph touches (bounces) at \( x=-4 \). The other options (7 and 4) are not roots (since \( (x - 7) \) or \( (x - 4) \) are not factors).

Answer:

\(-7\) (corresponding to the option with \(-7\))