at which root does the graph of f(x)=(x + 4)^6(x + 7)^5 cross the x - axis?\n-7\n-4\n4\n7

Explanation:

Step1: Recall the root - crossing rule

If a polynomial function (y = f(x)) has a factor ((x - a)^n), when (n) is odd, the graph of the function crosses the (x) - axis at (x=a), and when (n) is even, the graph of the function touches the (x) - axis at (x = a).

Step2: Find the roots of the function

For the function (f(x)=(x + 4)^6(x + 7)^5), set (f(x)=0). Then ((x + 4)^6(x + 7)^5=0). Using the zero - product property, we have (x+4 = 0) or (x + 7=0), so the roots are (x=-4) and (x=-7).

Step3: Analyze the exponents of the factors

The exponent of the factor ((x + 4)) is (n_1 = 6) (even), so the graph of (y = f(x)) touches the (x) - axis at (x=-4). The exponent of the factor ((x + 7)) is (n_2=5) (odd), so the graph of (y = f(x)) crosses the (x) - axis at (x=-7).

Answer:

-7