Question

Find the solution set of 2x^4-9x+11x^2-4x-6=0, if one of the roots is 1-i.

Answer 1

the other one is 1+1

Answer 2

so this thing factors as
\[ (x-(1+i))(x-(1-i))Q(x)\]

Answer 3

first two terms multiply to give
\[x^2-2x+2\]

Answer 4

so your annoying job is to divide
\[2x^4-9x^3+11x^2-4x-6\] by \[x^2-2x+2\]

Answer 5

first off is it clear that
\[(x-(1+i))(x-(i-i))=x^2-2x+2\]?

Answer 6

yessss :)

Answer 7

good. because it is easy once you know how to do it. -1-1 = -2 and 1 + 1 = 2

Answer 8

now you have a choice

Answer 9

you can use long division, or you can think

Answer 10

if you want to use long division i cannot show that here. but i can help you think through it if you like

Answer 11

would it work with synthetic?

Answer 12

nope

Answer 13

because you are not dividing by x - r

Answer 14

you are dividing by a quadratic.

Answer 15

so thinking is really the best method

Answer 16

ohhhh i see /:

Answer 17

i will start you out.
\[2x^4 -9x^3+11x^2-4x-6=(x^2-2x+2)(ax^2+bx+c)\]

Answer 18

now a and c are obvious yes?

Answer 19

yepp

Answer 20

name them

Answer 21

A is 2 and c is 11?

Answer 22

a is right, c is wrong. think about what the constant must be

Answer 23

is it -6?

Answer 24

nope but getting warmer. \[c\times 2=-6\]

Answer 25

that is the only way you get the constant from this product

Answer 26

so -3?

Answer 27

clear yes because all the other terms will have x's in them

Answer 28

so c must be -2 to give -6 as the constant and of course a = 2 so you get
\[2x^4\]

Answer 29

so you just factored it, right?

Answer 30

yes. now comes the only hard part, finding b
\[2x^4-9x^3+11x^2-4x-6=(x^2-2x+2)(2x^2+bx-3)\]

Answer 31

but we have many choices to find it, and it doesn't matter which method we pick

Answer 32

for example to get the "x" term it will come from \[2\times bx -3\times -2x\]
\[=(2b+6)x\] and we know that we have -4x so
\[2b+6=-4\]

Answer 33

i am just looking at the product and seeing what will give me an x term. we know we have to end up with -4x and if you multiply on the right you will have it

Answer 34

is it -5?

Answer 35

yes

Answer 36

we can also check with the \[x^3\] term

Answer 37

we know we have to end up with
\[-9x^3\] and that term will come from
\[bx^3-4x^3\]

Answer 38

so we know that
\[b-4=-9\] again telling us b = -5

Answer 39

oohh makes sense!!

Answer 40

it is more annoying to do it with the x^2 terms because there are several ways to get x^2

Answer 41

and of course you only have to check with one because (assuming your arithmetic is correct) it HAS to work because we know it factors.

Answer 42

i think this is easier than long division, but you can do it that way if you like. it is a mess. again the idea is to think what the first and last term has to be, and then find the middle one

Answer 43

like that myininaya?

Answer 44

thhats so much batter than long division. i just need to get this down. but thank you so much!

Answer 45

welcome

Answer 46

you can do polynomial division with more than just a linear term