Question

1. The volume is found by the function
f(x) = x3 + 3x2 – 10x – 24. You need to identify the correct factors of f(x).
Suggestions for the factors are:
(x – 1)
(x + 2)
(x – 3)
(x + 4)
(x + 6)
List the correct factors. Then justify your selections with complete sentences.

2. Professor Plum says that 2 is a zero of g(x) because long division with (x + 2) results in a remainder of 0. Miss Scarlett she says that 2 is a zero of g(x) because g(2) = 0.
Which guest is correct and which guest is incorrect? Explain your answer in complete sentences.

Answer 1

so have you plugged in 1,-2,3,-4,-6 into f to see which would give you 0 as an output

Answer 2

for example
f(1)=1^3+3(1)^2-10(1)-24
=1 +3 -10 -24
= 4 - 34=-30 which is not 0
so we know x=1 is not a zero so x-1 is not a factor

Answer 3

what happens when you plug in -2?

Answer 4

if f(n)=0, then x=n is a zero and x-n is a factor

Answer 5

(-2)^3=-8 not -2
(-2)^2=4 not 1
-10(-2)=20 not -10

Answer 6

\[f(-2)=(-2)^3+3(-2)^2-10(-2)-24 \\ \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ }=-8+3(4)+20-24 \\ \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } \text{ } =-8+12-4=4-4=0 \]

Answer 7

since f(-2)=0 then x=-2 is a zero and x+2 is a factor

Answer 8

f(3)=(3)^3+3(3)^2-10(3)-24
=27+3(9)-30-24
=27+27-20-24=10

Answer 9

you know you changed -30 to -20

Answer 10

f(3)=54-54=0

Answer 11

so that means x=3 is a zero and x-3 is therefore a factor

Answer 12

so you have f(x)=(x+2)(x-3)*something else

Answer 13

you need to find one more zero

Answer 14

and therefore one more factor

Answer 15

-4 or -6

Answer 16

plug into f to see which gives you 0

Answer 17

f(-4)=-4^3+3(-4)^2-10(-4)-24
=-64+48-40-24=-80 so no, x=-4 cant be a factor
that leaves -6, and as the last option....it has to be a factor

Answer 18

you are off by a sign in the expression -64+48-40-24

Answer 19

-10(-4)=40

Answer 20

-64+48+40-24=?

Answer 21

oh, wow....you are a life saver. thanks

Answer 22

it equals 0

Answer 23

right on!
hints for the question:
f(n)=0 means x=n is a zero and x-n is a factor
f(-n)=0 means x=-n is a zero and x+n is a factor
-
if we divide something function f(x) by x-r then we have
f(x)/(x-r)=q(x)+R/(x-r)
where R is the remainder and q(x) is the quotient
we can take this same equation and multiply both sides by (x-r)
f(x)=q(x)(x-r)+R
if we plug r into f we get f(r)=q(r)*(r-r)+R=q(r)*(0)+R=0+R=R
so R is the remainder when dividing f(x) by (x-r).
Now if R is a zero then x-r is a factor therfore x=r is a zero
---
we could have also chose to divide f by x+r
so if we did that then we have
f(x)=q(x)(x+r)+R
pluggining in -r gives f(-r)=R
if R again is 0, then x=-r is a zero then x+r is a factor

Answer 24

hints for the next question*