Question

Given f(x) = -2(x + 4)^9 (x + 3)^5 (x-2)^8, find the roots in increasing order.
The roots are , , and .
To the left of the first root, is the graph of f(x) above or below the x-axis? Answer above or below: .
Between the first two roots, is the graph of f(x) above or below the x-axis? Answer above or below: .
Between the last two roots, is the graph of f(x) above or below the x-axis? Answer above or below: .
After the last root, is the graph of f(x) above or below the x-axis? Answer above or below:

Answer 1

hey @minato ;D
so the roots of a function are when y = 0
(aka when does the function cross the x axis)
for something that's already factorised like your equation above, it's quite simple to calculate

Answer 2

Someone who recognized! Wooo! Anyways, go on xP

Answer 3

f(x) = -2(x + 4)^9 (x + 3)^5 (x-2)^8
roots are at f(x) = 0, so:
0 = -2(x + 4)^9 (x + 3)^5 (x-2)^8

now if you have 3 brackets, let's call them A , B , and C
f(x) = -2 ( A )^9 ( B )^5 ( C )^8

the possibilities for when f(x) = 0 will occur if:
A = 0
as
f(x) = -2 ( 0 )^9 ( B )^5 ( C )^8
f(x) = 0 ( B )^5 ( C )^8
and anything times 0 = 0, so it doesn't matter therefore what B and C are.

next possibility for f(x) = 0 will occur if:
B = 0
as
f(x) = -2 ( A )^9 ( 0 )^5 ( C )^8
f(x) = -2 ( A )^9 0 ( C )^8
and anything times 0 = 0, so it doesn't matter therefore what A and C are.

final possibility for f(x) = 0 will occur if:
C = 0
as
f(x) = -2 ( A )^9 ( B )^5 ( 0 )^8
f(x) = -2 ( A )^9 ( B )^5 0
and anything times 0 = 0, so it doesn't matter therefore what A and B are.

Answer 4

so first root:
A = 0
and
A = (x+4)
so
0 = x + 4
-4 = x

so f(x) = 0 if x = -4

next root:
B = 0
and
B = (x+3)
so
0 = x + 3
-3 = x

@hokage ;) this all makin sense so far dude?

Answer 5

and if so, are you right to solve for final root?

Answer 6

@Yondaime you still with me bro...?

Answer 7

Sorry about that!

Answer 8

I kind of get where you were going with this, but the above and below questions still have me at a loss.

Answer 9

that easy enough, there a simple trick to that too, but first, the final root
so C = 0
and bracket C = (x - 2)
so 0 = (x-2)
solve for x dude, then we'll do the above and below stuff, it'll take seconds

Answer 10

You mean X-2 <-- solve for x? Then x = 2?

Answer 11

perfect!
so your roots are
x = -4
x = -3
aaaaaaaaaaaand
x = 2

Answer 12

Alright, cool. I get that part. Thanks for everything so far by the way.

Answer 13

so those points... that's where a graph of your line will = 0
so coordinates are (-4,0) , (-3,0) and (2,0)

all good dude hey, naruto fans gotta stick together lol ;D

Answer 14

Lol, awesome. I like your DP btw ;D DBZ IS THE ROOT OF ALL OF IT.

Answer 15

now: above an below stuff --->
"To the left of the first root, is the graph of f(x) above or below the x-axis? "
the first root is (-4, 0 )
so to the left of that, we'll pick a point: how bout x = -5 ?
so when x = -5, f(x) = ...?

if f(x) = a negative number, it's below the line (x axis)
if f(x) = a positive number, it's above the line
if f(x) = 0, it's still on the line

so i'm gonna use an equation solver so we dont have to do the heavy maths here, but what you do is sub x = -5 into the big equation to get your answer for f(x)

SO:
f(x) = -2(x + 4)^9 (x + 3)^5 (x-2)^8 at x = -5
f(-5) = -2(-5 + 4)^9 (-5 + 3)^5 (-5-2)^8
= -2(-1)^9 (-2)^5 (-7)^8
= -368947264

which is a freaking huge NEGATIVE number, so f(x) is below the line

Answer 16

next one:
"Between the first two roots, is the graph of f(x) above or below the x-axis?"
so pick any number between x = -4 and x = -3 (lets go x = -3.5)

plug it into the equation as per the previous answer:
f(x) = -2(x + 4)^9 (x + 3)^5 (x-2)^8 at x = -3.5
f(-3.5) = -2(-3.5 + 4)^9 (-3.5 + 3)^5 (-3.5-2)^8

answer: f(-3.5) = 102.214

it's a positive number, so f(x) is ABOVE the line

(graph of the function attached)

Answer 17

next:
Between the last two roots, is the graph of f(x) above or below the x-axis?
so pick any number between x = -3 and x = -2 (lets go x = 0)

plug it into the equation as per the previous answer: (seeing a pattern here ;D )
f(x) = -2(x + 4)^9 (x + 3)^5 (x-2)^8 at x = 0
f(0) = -2(0 + 4)^9 (0 + 3)^5 (0-2)^8
f(0) = -2( 4)^9 ( 3)^5 (-2)^8

f(0) = -32614907904

massive number, but more importantly: NEGATIVE, so f(0) is BELOW the line

Answer 18

cool dude, now u go nuts on the last question:
"After the last root, is the graph of f(x) above or below the x-axis?"

Answer 19

so pick any number higher than x = 2

Answer 20

I see..wow, thanks a lot man. You took your time and really got it to me. Thank you so much.

Answer 21

all good dude, always welcome hey ;D

Answer 22

now you can either look at the graph or do the maths for the last Q, up to u
slaters ;D