one more using synthetic division please...

[alparnell]

Instructions are to show all 5 steps including synthetic divisions.
P(x)= x^8+2x^7-8x^6-13x^4-26x^3+116x^2+24x-96

I am cluess as to where to start as this looks like a really long answer to me. I missed a weekof class and won't be able to meet with my instructor until after an exam on this. I would greatly appreciate any help with this as I would really like to learn it for the upcoming test. Thank you so much for your help.

 

[HallsofIvy]

Instructions are to show all 5 steps including synthetic divisions.
P(x)= x^8+2x^7-8x^6-13x^4-26x^3+116x^2+24x-96

I am cluess as to where to start as this looks like a really long answer to me. I missed a weekof class and won't be able to meet with my instructor until after an exam on this. I would greatly appreciate any help with this as I would really like to learn it for the upcoming test. Thank you so much for your help.

I don't believe for a minute that the "Instructions are to show all 5 steps". "All 5 steps" to do what?? Normally "synthetic division" involves division but you need two polynomials to do that.

 

[soroban]

Hello, alparnell!

Instructions are to show all 5 steps including synthetic divisions. .??

. . \(\displaystyle P(x)\:=\: x^8+2x^7-8x^6-13x^4-26x^3+116x^2+24x-96\)

I would guess that the instructions are "Factor completely".
But synthetic division does not work for the entire problem.


\(\displaystyle \begin{array}{cccccccccccc} 1 &|& 1 & 2 & \text{-}8 & 0 & \text{-}13 & \text{-}26 & 116 & 24 & \text{-}96 \\
& | && 1 & 3 & \text{-}5 & \text{-}5 & \text{-}18 & \text{-}44 & 72 & 96 \\ & & -- & -- & -- & -- & -- & -- & --& -- & -- \\ && 1 & 3 & \text{-}5 & \text{-}5 & \text{-}18 & \text{-}44 & 72 & 96 & 0 \end{array}\)


\(\displaystyle \begin{array}{ccccccccccc}\text{-}1 & | & 1 & 3 & \text{-}5 & \text{-}5 & \text{-}18 & \text{-}44 & 72 & 96 \\ & | && \text{-}2 & \text{-}2 & 7 & \text{-}7 & 20 & 24 & \text{-}96 \\ && -- & -- & -- & -- & -- & -- & -- & -- \\ && 1 & 2 & \text{-}7 & 2 & \text{-}20 & \text{-}24 & 96 & 0 \end{array}\)


\(\displaystyle \begin{array}{ccccccccc} 2 & | & 1 & 2 & \text{-}7 & 2 & \text{-}20 & \text{-}24 & 96 \\ & | && 2 & 8 & 2 & 8 & \text{-}24 & \text{-}96 \\ && -- & -- & -- & -- & -- & -- & -- \\ && 1 & 4 & 1 & 4 & \text{-}12 & \text{-}48 & 0 \end{array}\)


\(\displaystyle \begin{array}{cccccccc}-4 &|& 1 & 4 & 1 & 4 & \text{-}12 & \text{-}48 \\ & | & & \text{-}4 & 0 & \text{-}4 & 0 & 48 \\ && -- & -- & -- & -- & -- & -- \\ && 1 & 0 & 1 & 0 & \text{-}12 & 0 \end{array}\)


So far, the factors are: .\(\displaystyle (x-1)(x+1)(x-2)(x+4)(x^4+x^2-12)\)

That trinomial cannot be factored by synthetic division.
(Well, it can be done . . . but I'd rather not.)

We have: .\(\displaystyle x^4 + x^2 - 12 \:=\x^2-3)(x^2+4)\)

. . which factors further: .\(\displaystyle (x - \sqrt{3})(x + \sqrt{3})(x - 2i)(x + 2i)\)

 

[alparnell]

I don't believe for a minute that the "Instructions are to show all 5 steps". "All 5 steps" to do what?? Normally "synthetic division" involves division but you need two polynomials to do that.

The teacher had hand written the intructions just as I had. I wasn't sure if I was supposed to do just as I had with the previous problem I posted or if this was completely different.
Math of any sort has never been a strong point for me and I sincerely do appreciate the replies.

 

[alparnell]

Soroban,
Thanks for posting that. I like long division better but your showing me how to do this helps tremendously! Very grateful!

 

[alparnell]

soroban...

Where did the -1, 2, and-4 come from? I had the 1 and did the synthetic division on that one but confused on the others. thanks.