at which root does the graph of $f(x)=(x - 5)^3(x + 2)^2$ touch the $x$-axis?\n-5\n-2\n2\n5
at which root does the graph of $f(x)=(x - 5)^3(x + 2)^2$ touch the $x$-axis?\n-5\n-2\n2\n5
Answer
Answer:
B. -2
Explanation:
Step1: Recall root - multiplicity rule
If (y = f(x)=(x - a)^n), when (n) is even, the graph of (y = f(x)) touches the (x) - axis at (x=a), and when (n) is odd, the graph of (y = f(x)) crosses the (x) - axis at (x = a).
Step2: Analyze the multiplicities
For the function (f(x)=(x - 5)^3(x + 2)^2), the root (x = 5) has multiplicity (n_1=3) (odd), so the graph crosses the (x) - axis at (x = 5). The root (x=-2) has multiplicity (n_2 = 2) (even), so the graph touches the (x) - axis at (x=-2).