Find all the real zeros of P

Question

A polynomial P is given.

P(x) = x³ − 5x² − x + 5

(a) Find all the real zeros of P. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)

x =

(b) Sketch the graph of P.

Answer 1

Hello Isaiah

First of all thanks to Mark above we are given the best and only lead needed to find your zeroes

You have the form

f(x) = ax³ + bx² + cx + d

The zeroes would simply be coefficients that yield zeroes

Since you have a 3rd degree polynomial you have 3 x intercepts and one y intercept

First the obvious one is the y intercept it occurs when x = 0

So for

f(x) = x³ - 5x² - x + 5

When x is 0, y =5

When y is zero you can simply evaluate your coefficients and your constant to find the x intercepts

in your function above

a =1

b = -5

c = -1

d = 5

Evaluate each for

0 = x³ - 5x² - x + 5

See if you get (a,0) (b,0) (c,0), and (d,0)

Since you know the value of a, b, c and d just plug them in one at a time.

And you know you have a 3rd degree polynomial so three zeroes,x intercepts, not four are expected.

Also you can graph your function at Desmos.com to confirm, see the zeroes answer part b and rea1ize Mark's lead for you above.

Give it a try.

Answer 2

bottom right graph is correct

with zeros +1, -1 and 5

factors: (x-1)(x+1)(x-5)