Question

Please help! Show me how to factor these binomials!
m^4-256 and z^3+125, thank you! Descriptions are appreciated :)

Answer 1

m^4-256
this is a perfect square

Answer 2

because 2*2=4, right?

Answer 3

the 1 you want to concentrate on is 256

Answer 4

ok

Answer 5

try and get it into the form \(a^2-b^2\)
Use the hints that mth3v4 has given you.

Answer 6

cool pic asna :D

Answer 7

thx :) I'm actually an alien from another planet :D

Answer 8

:)

Answer 9

Starshine7 - can you express 256 as (something)^2 ?

Answer 10

16x16?

Answer 11

:)

Answer 12

good, now can you express \(m^4\) as (something)^2 ?

Answer 13

so can you put that in in a factored form

Answer 14

m*m*m*m?

Answer 15

Yes, so \(m^4=(m^2)^2\)
Now use those two to get the expression into the form \(a^2-b^2\) so that you can then factor it like this:\[a^2-b^2=(a+b)(a-b)\]

Answer 16

why is it written like (m+number)(trinomial)? final answer i mean. If it helps, I factor by grouping if i can incorporate that somehow

Answer 17

just do this one step at a time and al will be revealed :)

Answer 18

ok

Answer 19

lol oops

Answer 20

nothing happened

Answer 21

:)

Answer 22

Starshine7 - are you stuck?

Answer 23

Sadly yes :(

Answer 24

look at the first answer you gave me

Answer 25

how did you come up with that ?

Answer 26

ok, we saw above that:\[256=16^2\]\[m^4=(m^2)^2\]so now we can rewrite your original equation as:\[m^4-256=(m^2)^2-16^2\]now mak use of:\[a^2-b^2=(a+b)(a-b)\]to simplify this further.

Answer 27

after that if can be factored again

as an option

as you can see

Answer 28

(m^2)^2 confuses me

Answer 29

(i think i am confusing starshine)

Answer 30

ok, lets use a substitution to make it clearer...

Answer 31

let \(n=m^2\), then we have:\[m^4-256=(m^2)^2-16^2=n^2-16^2\]

Answer 32

why n? o.O

Answer 33

we could use any other symbol - it doesn't really matter

Answer 34

the point is it is now in the form \(a^2-b^2\)

Answer 35

so you should now be able to factorise it.

Answer 36

okies

properties of exponents state

when you multiply exponent and exponent
you add the exponent values together

(a^x)(a^b)=a^x+b

Answer 37

exponent properties/laws of exponents
whatever

Answer 38

Starshine7 - are you familiar with the exponent laws mth3v4 has shown?
i.e.:\[(x^a)^b=x^{ab}\]

Answer 39

If it helps I dont understand when a and b are examples, I was taught using only the numbers, so thats why im really confused :(

Answer 40

those are just any number

Answer 41

I know >_> Im weird like that

Answer 42

so you are comfortable with something like this:\[18^2-16^2=(18+16)(18-16)\]but when replaced with symbols like this:\[a^2-b^2=(a+b)(a-b)\]you get lost?

Answer 43

Yes, exactly :)

Answer 44

ok - the 'a' and 'b' are just placeholders for any number

Answer 45

I just have attention issues so the numbers help me understand it better, i drift easily.

Answer 46

what this says is that the formula:\[a^2-b^2=(a+b)(a-b)\]is true for ANY value of 'a' and 'b'

Answer 47

so in your example above, we ended up with:\[m^4-256=(m^2)^2-16^2=n^2-16^2\]which means we can think of it as a=n and b=16 to get:\[n^2-16^2=(n+16)(n-16)\]we can then substitute back \(n=m^2\) to get:\[m^4-256=(n+16)(n-16)=(m^2+16)(m^2-16)\]

Answer 48

do you understand those steps?

Answer 49

I think so

Answer 50

(a ^x ) ( a ^b )= a^x+b
^ ^ ^ ^ ^ ^ ^
| | | | / | |
| L _____|___ L _ |___L_ _ L _ exponent number both x and b can be any number
| | |
your number ______ |

Answer 51

does my funny drawing help XD

Answer 52

ok, so now you should notice that we have \((m^2-16)\) as one of the factors. since \(16=4^2\), we can write this as:\[(m^2-16)=(m^2-4^2)=(m+4)(m-4)\]using the same formula.
we therefore end up with:\[m^4-256=(m^2+16)(m+4)(m-4)\]

Answer 53

So you just keep factoring to the lowest numbers that are perfect squares?

Answer 54

yes

Answer 55

mth3v4 - you have a lot of patience to draw all that :)

Answer 56

lol

Answer 57

:)

Answer 58

it still comes out a bit messy

Answer 59

(a^x) (a^y) = a ^ x+y

lemme try this

you have this thing here right starshine?

Answer 60

Starshine7 - for your second expression, you need to use the following factorisation:\[a^3+b^3=(a+b)(a^2-ab+b^2)\]so you need to get your expression:\[z^3+125\]into the form \(a^3+b^3\)

Answer 61

okies finish what you have first

Answer 62

I guess

Answer 63

I have to go now guys - Starshine I am sure mth3v4 will help you with the rest. Good luck all...

Answer 64

Aww..well bye Asna :) thanks anyways

Answer 65

kk bb

Answer 66

yw

Answer 67

z^3+125 is your next 1

Answer 68

Yes

Answer 69

it is not going to be squared
but cubed

Answer 70

have you heard of

diff of cubes
or sum of cubes

formulas

Answer 71

no

Answer 72

so what ansa wrote there before leaving was
formula for

diff of cubes

Answer 73

because you just see

just a 2 termed expression here

Answer 74

so how would you make

x^3
and change it to
x^2

Answer 75

better yet lemme ask the question another way

how do you get
x^3
from
x^2

Answer 76

I'm not sure. I think they only have 2 exs in common

Answer 77

okies
lets try this

( a^x ) ( a^y ) = a ^ x +y

the variable
"a"
can be any number 1, 2, 3, whatever, happy face , i dont care what you want it to be
ok so far

Answer 78

this just goes fot the variable

"a"

we good?

Answer 79

yes

Answer 80

then next

this for the variables

"x, y"

these give a numeric value (these have to be a number)

giving a value

( a^x ) ( a^y) = a ^ x + y

Answer 81

so in the end

the numeric value of your

"a" exponent value "x"

"a" exponent value "y"

they add up

result for the a ^ x + y

is this more understandable?

Answer 82

Yes

Answer 83

so can we try that again

how to get
x^3
from
x^2

Answer 84

(x)(x+x)?

Answer 85

i just want it from x^2

Answer 86

I'm sorry I have no idea what to do :(

Answer 87

since it is already squared right

(x^2)(x)=x

how many more "x"

do we need for x ^3

Answer 88

x^2

Answer 89

you have x^2 already

Answer 90

but x^3 is made up of x^3 theres not anything left to do

Answer 91

remember

(x^2)(x)=x?
^
|
value of exponent you are adding to the result

Answer 92

lol
if there is nothing then say nothing

Answer 93

be more sure of you self
now just give the reason

Answer 94

nm that its probably more difficult to explain

Answer 95

but do you understand how it works

Answer 96

thats all i want to know

Answer 97

How about we start over with a new problem, m^3+216n^3 Show me in steps how to solve this and ill try another on my own, this might work

Answer 98

okeedokee