how to solve these questions

Question

Find all the zeros of the following polynomial function, rewrite it in factored form, and then sketch

its graph.

𝑔(𝑥)=2𝑥3 +5𝑥2 −32𝑥−80

Use the rational zeros theorem to LIST the potential rational zeros of the given polynomial function. Then find all the zeros (real or otherwise) of 𝑝(𝑥). 𝑝(𝑥) =𝑥4 +6𝑥3 +10𝑥2 −2𝑥−15

Determine the domain, asymptotes, location of the hole and all intercepts of the following rational function. Then sketch the graph of 𝑅(𝑥). 𝑅(𝑥) =−2(𝑥−3)(𝑥+3)(𝑥+4) (𝑥−5)(𝑥+3)(𝑥+1)

Answer 1

2x^3 +5x^2 -32x -80, factor by grouping

= x^2(2x +5) -16(2x+5)

= (x^2-16)(2x+5)

= (x+4)(x-4)(2x+5)

roots = zeros = the x intercepts = 4,-4, and -2.5

y intercept = -80

for R(x) there are 6 x intercepts -1, -2, -3, -3, 3, 5. one repeats as the curve is tangent to the x axis when x=-3

Answer 2

Use grouping

g(x) = 2x³ - 32x + 5x² - 80

For p(x) use the Rational Root Theorem (as suggested) then Synthetic Substitution

As written R(x) has a domain of all Real numbers. It does not have any holes or asymptotes.