anonymous
  • anonymous
How do you use the given root to assist in finding the remaining roots of the equation? 2x^4 - 15x^3 + 34x^2 - 19x - 20 = 0; 1/2
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
divide. if \[\frac{1}{2}\] is a root then you know your polynomial is \[(x-\frac{1}{2})q(x)\] or more easily written as \[p(x)=(2x-1)q(x)\]
anonymous
  • anonymous
you find \[q(x)\] by dividing by \[(x-\frac{1}{2})\] synthetic division is easiest
anonymous
  • anonymous
do you know how to use synthetic division?

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mathteacher1729
  • mathteacher1729
These are the synthetic division steps. You know to use synthetic division because you are given a factor of polynomial. It's like saying "458 is divisible by 2... what is the remainder when divisible by 2?) Did you copy your polynomial correctly? You are left with a third degree polynomial which has complex roots... Usually the problems give "Friendlier" answers. :)
anonymous
  • anonymous
if this is correct there must be a typo somewhere because you (wolfram) got a remainder of -91/4
anonymous
  • anonymous
Great
anonymous
  • anonymous
remainder should be 0 of course
mathteacher1729
  • mathteacher1729
Jennifer -- I think you might have mis-copied the problem. The answer is ... too ugly for something which might be assigned for a practice/hw or even test problem.
anonymous
  • anonymous
Let me check quick
anonymous
  • anonymous
in any case 1/2 is not a zero according to wolfram
anonymous
  • anonymous
Oh wow, after the equation, the 1/2 is a negative..I am so sorry..
anonymous
  • anonymous
2x^4 - 15x^3 + 34x^2 - 19x - 20 = 0; -1/2
anonymous
  • anonymous
ohhhhhhhhhhhhhhhhhhhh -1/2
anonymous
  • anonymous
You sure are one smart cookie to notice that. My goodness; i sure wish I could get math as easy as you .
mathteacher1729
  • mathteacher1729
That makes more sense, but you're still going to be left with a 3rd degree polynomial... And that 3rd order polynomial will not be very easy to solve.
anonymous
  • anonymous
then it factors as \[(x+\frac{1}{2})q(x)\]
mathteacher1729
  • mathteacher1729
anonymous
  • anonymous
btw you can find the quotient relatively easily. very easy if you can synthetic division
anonymous
  • anonymous
do you know how? long division is a pain.
anonymous
  • anonymous
a little bit..i'm bad..really
anonymous
  • anonymous
list the coefficients they are 2 -15 34 -19 20
anonymous
  • anonymous
since you root is -1/2 put that on the side. i will write -.5 2 -15 34 -19 20 -.5 ___________________________________
anonymous
  • anonymous
drop the 2 2 -15 34 -19 20 -.5 ___________________________________ 2
anonymous
  • anonymous
-.5*-1 2 -15 34 -19 20 -.5 -1 ___________________________________ 2
anonymous
  • anonymous
-15-1=-16 2 -15 34 -19 20 -.5 -1 ___________________________________ 2 -16
anonymous
  • anonymous
-16*-.5 = 8 2 -15 34 -19 20 -.5 -1 8 ___________________________________ 2 -16
anonymous
  • anonymous
34 + 8 = 42 2 -15 34 -19 20 -.5 -1 8 ___________________________________ 2 -16 42
anonymous
  • anonymous
-.5*42=-21 2 -15 34 -19 20 -.5 -1 8 -21 ___________________________________ 2 -16 42
anonymous
  • anonymous
-19-21=-40 2 -15 34 -19 20 -.5 -1 8 -21 ___________________________________ 2 -16 42 -40
anonymous
  • anonymous
-.5*-40= 20 and now i see i have a typo. the last number in the top line should be -20, not 20 easy to fix 2 -15 34 -19 - 20 -.5 -1 8 -21 20 ___________________________________ 2 -16 42 -40
anonymous
  • anonymous
-20+20=0 so the remainder is 0 \ 2 -15 34 -19 - 20 -.5 -1 8 -21 20 ___________________________________ 2 -16 42 -40 0
anonymous
  • anonymous
and your answer is \[p(x)=(x+\frac{1}{2})(2x^3-16x^2+42x-40)\]
anonymous
  • anonymous
or \[p(x)=(2x-1)(x^3-8x^2+21x-10)\]
anonymous
  • anonymous
done!

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