Question

1.A mysterious box is delivered to the dinner party you are attending. The label on the box says that the volume of a box is the function f(x) = x^3 + 3x^2 – 10x – 24. To open the box, you need to identify the correct factors of f(x). Party-goers offer up solutions, and it is your job to find the right ones.


Their suggestions are:
•(x – 1)
•(x + 2)
•(x – 3)
•(x + 4)
•(x + 6)
•(x – 12)


List the correct factors. Then justify your selections with complete sentences.

Answer 1

@austinL

Answer 2

You can use synthetic division to do this

Answer 3

although i don't even know how to do the box in the left on word for it.. does anyone know

Answer 4

just use the factor theorem

which means, substitute a possible zero into the polynomial, when simplified, if the answer is zero then you have a zero and hence factor of the polynomial.

as an example look at the possible factor

(x + 2) which means x = -2 is a possible zero.

then you need to find f(-2) and see if its zero.

so

\[f(-2) = (-2)^3 + 3\times (-2)^2 -10 \times(-2) - 24 = -8 + 12 + 20 -24\]

and when you evalute it you find

f(-2) = 0

so then you can say (x + 2) is a factor of the polynomial.

here is a link to a site that explains the factor theorem

hope this helps

Answer 5

ohh, but how come when I just tried it with synthetic division, it wasn't a factor..?

Answer 6

for (x-2)

Answer 7

and btw u didn't put a link^^

Answer 8

well I showed (x + 2) is a factor.... not (x -2)

and I showed it was a factor by substituting x = -2 into the cubic.

here is another clue

look at the cubic

f(x) = (x +2)(x +a)(a + b)

the product of the 3 roots and the constant term in the cubic

\[2 \times a \times b = -24\]

so you are looking for a pair of factors where the constants multiply to -12..

so it will eliminate a few pairs.

Answer 9

it was a typo

Answer 10

thanks so much

Answer 11

so which factor do u think i should test next

Answer 12

well I would say not (x -1)

so I'd say (x-3) and then x + 4) perhaps

Answer 13

so substitute

f(3) then f(-4) and see if they equal zero

Answer 14

ok

Answer 15

@campbell_st

Answer 16

help.. can you explain what everything in the formula means f(x) = (x +2)(x +a)(a + b) the a, b, and x

Answer 17

im doing (x+3) now. So I know -3 will be x, but what's a and b?

Answer 18

oh wait.. i just plug it in to the original equation right

Answer 19

Idk if I did it right.. but i just tried it and i got -1224 ...

Answer 20

well you know (x + 2) is a factor

now text (x -3) by substituting x = 3 into the equation if its equal to zero you'll have another factor.

then f(x) = (x +2)(x -3) and the last one should be (x + 4)

to explain say something like,

by using the factor theorem, which requires the possible zeros of the cubic to be substituted, I found f(-2) = 0

which meant (x + 2) is a factor....

then tested f(3) and found it was equal to zero so (x -3) was a factor

so f(x) = (x +2)(x -3)(x + a)

the product of 2, -3, and a has to equal -24

so set up an equation -6a = -24

and solve for a a = 4

so the factors are

f(x) = (x +2)(x -3)(x + 4)

hope this helps

Answer 21

wait!

Answer 22

where did the " f(x) = (x +2)(x -3)" come from?

Answer 23

wait.. r u sure this is called the factor theorem? cuz this doesn't look like it: http://www.purplemath.com/modules/factrthm.htm

Answer 24

well is x +2 is a factor then x = -2 is a zero

and (x -3) is a factor then x = 3 is a zero


f(-2) = 0 is shown above

so look at f(-3)

\[f(-3) = (3)^3 + 3(3)^2 -10(3) - 24 = 27 + 27 - 30 - 24 = 0\]

so x = 3 is a zero and hence (x - 3) is a factor

so now you have 2 factors

f(x) = (x +2)(x -3)

just find the last

Answer 25

i keep doing it wrong

Answer 26

well if you read the 2 examples in the link...purplemath

they used synthetic division to so.... x = -4 is a zero and hence x + 4 is a factor in example 2. in example 1 the are asking to test if x - 1 is a factor, so they substituted the zero x = 1 and found it wasn't a factor... then supported the claim with synthetic division.

and it is the same...

Answer 27

ok i think i get it now thanks i gtg

Answer 28

so if you look at the quadratic

\[f(x) = x^2 - x - 6\]

it can be factored to

\[f(x) = (x -3)(x + 2)\]

the equation has been rewriten in factored form

so to find the values of x that make the equation equal zero you need to solve

x - 3 = 0 therefore x = 3

x + 2 = 0 therefore x = -2

either value will make the equation equal zero..