which of the following describes the zeroes of the graph of $f(x)=3x^{6}+30x^{5}+75x^{4}$?\n-5 with…

which of the following describes the zeroes of the graph of $f(x)=3x^{6}+30x^{5}+75x^{4}$?\n-5 with multiplicity 2 and $\frac{1}{3}$ with multiplicity 4\n5 with multiplicity 2 and $\frac{1}{3}$ with multiplicity 4\n-5 with multiplicity 2 and 0 with multiplicity 4\n5 with multiplicity 2 and 0 with multiplicity 4

which of the following describes the zeroes of the graph of $f(x)=3x^{6}+30x^{5}+75x^{4}$?\n-5 with multiplicity 2 and $\frac{1}{3}$ with multiplicity 4\n5 with multiplicity 2 and $\frac{1}{3}$ with multiplicity 4\n-5 with multiplicity 2 and 0 with multiplicity 4\n5 with multiplicity 2 and 0 with multiplicity 4

Answer

Answer:

C. -5 with multiplicity 2 and 0 with multiplicity 4

Explanation:

Step1: Factor the polynomial

[ \begin{align*} f(x)&=3x^{6}+30x^{5}+75x^{4}\ &=3x^{4}(x^{2} + 10x+25)\ &=3x^{4}(x + 5)^{2} \end{align*} ]

Step2: Find the zeros

Set (f(x)=0). Then (3x^{4}(x + 5)^{2}=0). Since (3\neq0), we have (x^{4}=0) or ((x + 5)^{2}=0). For (x^{4}=0), (x = 0) with multiplicity 4 (because the exponent of (x) is 4). For ((x + 5)^{2}=0), (x=-5) with multiplicity 2 (because the exponent of ((x + 5)) is 2).