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Question 1081688: Given the cubic function g(x)=x^3 + 6x +20. Show that g(x) can be written as g(x)=(x+2)Q(x), where Q(x) is a real quadratic. Hence,solve the equation g(x)=0 completely.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! see the following worksheet.
step 1 copies the original equation that will be divided by x+2.
step 2 inserts the missing order of exponents, so they're all present.
what was missing was the x^2 term, which was inserted as 0 * x^2.
step 3 justtakes the coefficients and divides they by -2 using the concept of synthetic division.
you are dividing by x + 2.
set x + 2 = 0 and solve for x.
you will get x = -2
that's what you divide by.
you bring down the first coefficient.
you then multiply that by -2 (not divide).
the result is then added to the next coefficient.
in this case the 1 was multiplied by -2 and then added to 0 to get -2.
you repeat that down the line until you get to the last term.
if that term is 0, then x+2 is a factor of x^3 + 6x + 20.
it is.
step 4 then adds the variable back in, but the order is one less than you started with.
that gets you x^2 -2x + 10.
step 5 takes (x+2) and multiplies it by (x^2 - 2x + 10)
this is to confirm the solution is correct.
step 6 completes the two parts of the multiplicsation and adds them together.
the result is what you see in step 7.
this is the original equation.
this tells you that the synthetic division was successful.
becauswe the last result in step 3 was 0, that told you that (x+2) divides evenly into x^3 + 6x + 20, which tells you that (x+2) is a factor of x^3 + 6x + 20.
here's a reference on synthetic division.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/col_algebra/col_alg_tut37_syndiv.htm
you could also have used long division and gotten the same result.
if the remainder was 0, then x+2 could be considered a factor of x^3 + 6x + 20
here's a reference on long division.
http://www.wtamu.edu/academic/anns/mps/math/mathlab/int_algebra/int_alg_tut35_div.htm
if the remainder was 0, then x+2 could be considered a factor of x^3 + 6x + 20
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