Question

f(x)= x(x-3)^2

State the degree and list the zeroes of the polynomial function. State the multiplicity of each zero and whether the graph crosses the x axis at the corresponding x intercept. Then sketch the graph of the polynomial function by hand.

Answer 1

If you expand the function you get:\[f(x)=x^3-6x^2+9x\]That means the dgree is 3.
From the first formula you wrote, we can seee that or x=0 or x-3=0 for the function to be 0. Then x=0 or x=3 makes the function 0. Now, the multiplicity is the dgree of the term that is 0 when x is a zero of the function. when x=0, the degree is 1, and it has multiplicity 1. if x=3, the term has degree 2, and the multiplicity is 2. If the zero has multiplicity 1, then the function is a line multiplied by something, and a line crosses the x axis. If it has multiplicity 2, it is a parabola multiplied by x, and since the given parabola do not cross, but touch the x axis, and the function is aroud 3 at that point, so it does not cross.

Answer 2

Thanks!