f(x)=(3x + 2)^6(x^2 - 16x + 64)\nx = (smallest x - value) with multiplicity\nx = (largest x - value) with…

Explanation:

Step1: Factor the quadratic part

We know that (x^{2}-16x + 64=(x - 8)^{2}) by the perfect - square formula ((a - b)^2=a^{2}-2ab + b^{2}) where (a = x) and (b = 8). So (f(x)=(3x + 2)^{6}(x - 8)^{2}).

Step2: Find the roots

Set (f(x)=0). Then ((3x + 2)^{6}(x - 8)^{2}=0). Using the zero - product property, if (ab = 0), then (a = 0) or (b = 0). For ((3x+2)^{6}=0), we solve (3x+2 = 0), which gives (x=-\frac{2}{3}). For ((x - 8)^{2}=0), we solve (x - 8=0), which gives (x = 8).

Step3: Determine the multiplicities

The multiplicity of a root (r) of a polynomial (P(x)) is the exponent of the factor ((x - r)) in the factored form of (P(x)). The root (x=-\frac{2}{3}) comes from the factor ((3x + 2)^{6}), so its multiplicity is (6). The root (x = 8) comes from the factor ((x - 8)^{2}), so its multiplicity is (2).

Answer:

(x=-\frac{2}{3}) (smallest x - value) with multiplicity (6) (x = 8) (largest x - value) with multiplicity (2)