Question

Find the zeros for the following polynomial function and give the multiplicity for each zero state whether the graph crosses the x-axis or touches and turns around... f(x)=9(x+8)(x+6)^3
What is the value of the zero with the smaller multiplicity?

Answer 1

well, if you set the equation to 0
that is \(\bf 0=9(x+8)(x+6)^3\)
the roots are really obvious
\(\bf0= 9(x+8)(x+6)^3 \implies 0 =9(x+8)(x+6)(x+6)(x+6)\)

Answer 2

now you have really 2 non-repeated roots
the one raised to 3, is really the root with the multiplicity of 3
so the multiplicity will be the exponent of the binomial for that root
so the other will have the "smaller" multiplicity

Answer 3

does it cross or just touch the x-axis?
well, keep in mind that, if the multiplicity, the exponent, is EVEN, it "touches" only, if it's ODD, it "crosses"