Algebra 3-4

[dense nexus]

There is a problem on my homework I can not figure out.

Graph the following polynomial. SHOW ALL WORK.
f(x) = -(3x - 15)(2x - 2)^3(x + 3)^2

I honestly don’t even know how to start

[plucky elk]

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[sweet furnace]

Find its zeroes and intervals of constant sign first.

As usual, you can just follow the general plan.

1. Domain.
2. Symmetry (evenness/oddness).
3. Periodicity, antiperiodicity.
4. Discontinuities, behavior around them.
5. Behavior at infinity, asymptotes.
6. x- and y-intercepts.
7. Intervals of constant sign.
8. Intervals of monotonicity, extrema.
9. Intervals of convexity, inflection points.
10. Range.

First four points are pretty much answered immediately for a polynomial.

[torpid elk]

note that roots that have odd multiplicity (that is, they have factors in the form (x-r)^(2k+1) where k is a natural number) passes straight though the x-axis, while roots that have an even multiplicity (i.e., factors in the form (x-r)^(2k) where k is a natural number) will “bounce back” or act like a parabola on its vertex in the local neighborhood (around) that root

[hollow tartan]

Solve 3x - 15 = 0, 2x - 2 = 0, x + 3 = 0 for roots. Odd = cross, even = bounce. Degree 6, negative = up left, down right. Plug x = 0 for y-intercept.

I honestly am bored it's 12 am

[torpid elk]

i think this has been covered by both comments.

[misty monolith]

@dense nexus

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