. The volume in cubic feet of a box can be expressed as (x)=x^3-〖6x〗^2+8x , or as the product of three linear factors with integer coefficients. The width of the box is 2 – x.

Factor the polynomial to find linear expressions for the height and the length

Question

. The volume in cubic feet of a box can be expressed as (x)=x^3-〖6x〗^2+8x , or as the product of three linear factors with integer coefficients. The width of the box is 2 – x.

Factor the polynomial to find linear expressions for the height and the length

jim_thompson5910 · Answer

what is the GCF of:  x^3, -6x^2, 8x

jim_thompson5910 · Answer

wait, is it (6x)^2 or 6x^2 ?

yanasidlinskiy · Answer

Let me check.

yanasidlinskiy · Answer

Sorry..$$-6x^2$$

jim_thompson5910 · Answer

that's fine

jim_thompson5910 · Answer

what is the GCF of: x^3, -6x^2, 8x ?

yanasidlinskiy · Answer

Anyways...Would it be 2?

jim_thompson5910 · Answer

not quite

yanasidlinskiy · Answer

I don't know then..I was thinking 2. The entire time.

yanasidlinskiy · Answer

How come it's not 2?

jim_thompson5910 · Answer

because 2 isn't found in the coefficient of x^3

jim_thompson5910 · Answer

what factor is in all three terms: x^3, -6x^2, 8x ?

yanasidlinskiy · Answer

Oh..Ok. x? I really don't knowxD

jim_thompson5910 · Answer

yep x

yanasidlinskiy · Answer

Hahha:D Wow.

jim_thompson5910 · Answer

that means 

x^3-6x^2+8x = x(x^2 - 6x + 8)

jim_thompson5910 · Answer

now factor x^2 - 6x + 8 to get ???

yanasidlinskiy · Answer

Umm. x^2 - 6x + 8 = x(x^2 - 6x + 8). Right?

jim_thompson5910 · Answer

To factor x^2 - 6x + 8, you need to find two numbers that

multiply to 8 (third term)

AND

add to -6 (second coefficient)

yanasidlinskiy · Answer

Ok. Can you show me step by step how? I just got into this class and I don't have a clue.

yanasidlinskiy · Answer

Oh, wait. Would it be 2?

jim_thompson5910 · Answer

what are some ways to factor 8?

yanasidlinskiy · Answer

Well 2*4 = 8.

yanasidlinskiy · Answer

(2*-4)(2*4) <---Is that right?

jim_thompson5910 · Answer

notice how 2+4 = 6

jim_thompson5910 · Answer

we want the two factors to add to -6, not +6

jim_thompson5910 · Answer

we can easily fix this by saying

(-2)*(-4) = 8

-2 + (-4) = -6

jim_thompson5910 · Answer

therefore,

x^2 - 6x + 8 = (x-2)(x-4)

yanasidlinskiy · Answer

And that would be my final answer?

jim_thompson5910 · Answer

so we have this full factorization

x^3-6x^2+8x = x(x-2)(x-4)

jim_thompson5910 · Answer

it says "The width of the box is 2 – x. "

notice how we have x-2 and not 2-x as a factor

jim_thompson5910 · Answer

we can easily fix this by pulling out a negative from each x-2 and x-4 

x-2 = -1(-x+2) = -1(2-x)

x-4 = -1(-x+4) = -1(4-x)

jim_thompson5910 · Answer

so,

x^3-6x^2+8x = x(x-2)(x-4)

x^3-6x^2+8x = x( -1(2-x)  )( -1(4-x)  )

x^3-6x^2+8x = x*( -1)*(-1)(2-x)(4-x)

x^3-6x^2+8x = x*(1)(2-x)(4-x)

x^3-6x^2+8x = x(2-x)(4-x)

yanasidlinskiy · Answer

Wow! That's a lot. And..That makes more sense. Would that be all?

jim_thompson5910 · Answer

|dw:1406079532516:dw|

jim_thompson5910 · Answer

so that means the product of the length and height is x(4-x)

jim_thompson5910 · Answer

the length could be x, or it could be 4-x

the height could be x, or it could be 4-x

jim_thompson5910 · Answer

it really doesn't matter since you can multiply in any order

yanasidlinskiy · Answer

Oh, wow. My teacher like "preached" to me that It does matter. She made a lot of mistakes anyways. So I'll rather trust you more than her:D

jim_thompson5910 · Answer

here is an example

2*4 = 8
4*2 = 8

jim_thompson5910 · Answer

multiplying can be done in any order

yanasidlinskiy · Answer

Exactly!!:D I tried to tell her and she's like..No, no no. You're wrong. And I'm like...Whatever.

jim_thompson5910 · Answer

strange, that's basic math that any teacher would know

yanasidlinskiy · Answer

Hahha:D Yep! I don't even know how she got hired. Lol.

jim_thompson5910 · Answer

A strange reality in education, yeah I know