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 crossing rule

A polynomial graph crosses the \(x\)-axis at a root if the multiplicity of that root is odd. If the multiplicity is even, the graph touches the \(x\)-axis (doesn't cross).

Step2: Find roots and their multiplicities

For \(f(x)=(x + 4)^{6}(x + 7)^{5}\), set each factor to zero:

• For \((x + 4)^{6}=0\), root is \(x=-4\) with multiplicity \(6\) (even).
• For \((x + 7)^{5}=0\), root is \(x=-7\) with multiplicity \(5\) (odd).

Since multiplicity of \(-7\) is odd, the graph crosses the \(x\)-axis at \(x = -7\).

Answer:

\(-7\)