Question

Explain, in complete sentences, how you would completely factor 3x3 + 27x2y - 24xy - 216y2 and check your factors for accuracy.

If someone could please explain this to me, that would be great because I'd like to be able to learn how to do this on my own. :)

Answer 1

let's first group it in 2 pairs,
\(\bf (3x^3 + 27x^2y) - (24xy + 216y^2)\)

can you get common factors for each of those pairs?

Answer 2

3x3+27x2y-24xy-216y2=
3x2(x+9y)-24y(x+9y)=
(3x2-24y) (x+9y)=
3(x2-8y) (x+9y)

Answer 3

For the first pair, would the common factor be three?

Answer 4

yes

Answer 5

well, a bit more than 3

Answer 6

Okay, so would I have to divide the 27x2y by three?

Answer 7

3x3 + 27x2y - 24xy - 216y2
First pair will be 3x^2(3+9y) and second will be -24(x+9y)
So,
(x+9y)(3x^2-24)

Answer 8

-24y*

Answer 9

I don't get it, I need an explanation, like how do you find those numbers? :(

Answer 10

Hm what's common in 3x^3 +27x^2y? <--- 3 x 1= 3 and 3 x 9=27

Answer 11

Get it till here?

Answer 12

I understand that 3 and 27 have three in common but where does the (3+9y) come from? Because the 27 has an x^2 as well....

Answer 13

$$
3x^3 + 27x^2y - 24xy - 216y^2 \implies
(3x^3 + 27x^2y) - (24xy + 216y^2)\\
\color{green}{\text{now we get common factors on both}}\\
3x^2\color{blue}{(x + 9y)} - 24y\color{blue}{(x + 9y)}\\
\color{green}{\text{now we see 2 common factors and we take them out}}\\
(3x^2-24y)(x + 9y)\\
$$

Answer 14

I don't understand.... :(

Answer 15

@Breeziee Try asking your teacher?

Answer 16

She isn't online right now. I just don't understand how you get the x+9y thing is that by dividing the 27 thing by three

Answer 17

We take the common from
3x^3+27x^2y
3x^2(x+9y) .....3x^2 is common!!

Answer 18

\(\bf (3x^3 + 27x^2y) \implies (3xxx + 3 \times 9xxy)\)

common factor, see what's common to both?

Answer 19

Yes, I see it now

Answer 20

Thankgod :P

Answer 21

Sorry, I'm just dumb. :(

So what about the second half of the equation?

Answer 22

24xy - 216y2

24 x1 =24 and 24x9 =126

In this equation 24xy has one y and 216 has 2 y's how many are common? 1 y
So 24y(x+9y)

Answer 23

216* not 126..sorry typo.

Answer 24

Okay, I get that. Now what's next?

Answer 25

\(\bf (24xy + 216y^2) \implies (24xy + 24\times 24yy)\)
common factor there too

Answer 26

\(3x^3 + 27x^2y - 24xy - 216y^2 \implies
(3x^3 + 27x^2y) - (24xy + 216y^2)\\
\color{green}{\text{now we get common factors on both}}\\
3x^2\color{blue}{(x + 9y)} - 24y\color{blue}{(x + 9y)}\\
\color{green}{\text{now we see 2 common factors and we take them out}}\\
(3x^2-24y)(x + 9y)\)

Answer 27

The just that,
3x^2(x+9y) -24y(x+9y)
(x+9y) is same so we write it only one time and 3x^2-24y
(x+9y)(3x^2-24y)

Answer 28

Oh, thank you so much, I think I get it now!

Answer 29

Yw :)

Answer 30

Thank you so much though for putting up with me, I can be a little difficult sometimes. :)