Question

factor the expression and use fundamental identities to simplify.
1- 2cos^2x+cos^4x

Answer 1

you can rewrite it as \[1 - 2 (\cos x)^2 + (\cos x)^4\]

Answer 2

now watch the magic..

let a = cos x

1 - 2a^2 + a^4

let b = a^2

1 - 2b + b^2

Answer 3

are you following? or did i lose you?

Answer 4

sorry sorry xD

Answer 5

so why are we replacing it with variables? :o

Answer 6

to make it look nicer and easier

Answer 7

you can just use cos

Answer 8

i substituted so you can see it clearly

Answer 9

shall i continue?

Answer 10

oh okay got it, go on

Answer 11

haha sorry x)

Answer 12

ahh good

1 - 2b + b^2

im going to rearrange it to make it look nicer

b^2 - 2b + 1

now you can see this is factorable

(b-1)(b-1) <--i hope you saw that

Answer 13

now we sub b back to a

b = a^2 so

\[(a^2 - 1)(a^2 + 1)\]
the first factor is still factorable so let's

\[(a+1)(a-1)(a^2 + 1)\]
now we sub a back to cosx

a = cosx
\[(\cos x + 1)(\cos x - 1)(\cos^2 x + 1)\]

Answer 14

i hope you got my explanation. sorry for rushing. it's just that im about to leave lol

Answer 15

awww :( i was just gonna say i got lost when you factored lol

Answer 16

uhmm which part?

Answer 17

b^2 - 2b + 1 = (b-1)(b-1)?

Answer 18

oh no...my last post was wrong o.O

do you get (b-1)(b-1) though?

Answer 19

yeah that part

Answer 20

it's a perfect square trinomial

Answer 21

i dont know how to teach factoring.....it's a little complicated to explain...

Answer 22

this is gonna be a little tricky...the next step would require a more advanced factoring...

Answer 23

let me post the solution first then i'll tag someone to help explain it hehe...i need to go..

Answer 24

okay sorry about that! but thanks for the help :)

Answer 25

(b-1)(b-1)
sub back to b = a^2

(a^2 - 1)(a^2 - 1)
factor out a negative

-(1 - a^2)(a^2 - 1)
factor out another negative

(1 - a^2)(1-a^2)

sub a = cosx

(1 - cos^2 x)( 1 - cos^2 x)

(sin^2 x) (sin^2 x)

sin^4 x

Answer 26

@angela210793 could you help me explain the solution?

i need to go :(

Answer 27

sorry again @dorkkk