How do you use synthetic division to divide #2x^3-3x^2-5x-12div x-3#?

2 Answers
Feb 25, 2016

Answer:

#2x^2+3x+4#

Explanation:

The first thing we must do is find the value that makes #x-3# equal to zero. In this case, #x=3#. This will be our divisor.

Now, we set up the problem:
If we have #color(green)(2)x^3color(red)(-3)x^2color(blue)(-5)xcolor(purple)(-12)#

#color(white)(3.)##|##color(green)(2)color(white)(3.)color(red)(-3)color(white)(3.)color(blue)(-5)color(white)(3.)color(purple)(-12)#
3#color(white)(.)##|#______

Now, we bring down the #color(green)(2)# and multiply it by the #3#, like this:
#color(white)(3.)##|##color(green)(2)color(white)(3.)color(red)(-3)color(white)(3.)color(blue)(-5)color(white)(3.)color(purple)(-12)#
3#color(white)(.)##|#__#6#____
#color(white)(.......)color(green)(2)#

Now we add the #color(red)(-3)# to the #6#, which gives us #3#. I'll show you what I mean.

#color(white)(3.)##|##color(green)(2)color(white)(3.)color(red)(-3)color(white)(3.)color(blue)(-5)color(white)(3.)color(purple)(-12)#
3#color(white)(.)##|#__#6#___#9#_#12#__
#color(white)(.......)color(green)(2)##color(white)(....)3##color(white)(.......)4##color(white)(.......)0#

I just did the whole thing, and I hope you can see what I did. I took the sum of the answer and multiplied it to the divisor, #3#. Then I take the product and place it in the next column. Then, I add the column together, and whatever the answer is, I multiply it to the #3#.

Anyways, we take the leftover number as the bottom, the #color(green)(2)color(white)(.)3 color(white)(.)4 color(white)(.)0# and rewrite them as an equation, like this: #2x^2+3x+4#. We can factor this further if we want, but I'm going to stop there.

Feb 25, 2016

Answer:

#(2x^3-3x^2-5x-12) -:( x-3) = 2x^2+3x+4#

Explanation:

This answer uses polynomial long division - for synthetic division, please see the other answer!

We write the polynomial and its divisor down in long division form and work through the normal steps of long division:

Then we guess our first term in the quotient, which should subtract from the first term in the dividend:

enter image source here

After performing the subraction, we a left with our first remainder. We next drop the remaining terms in the dividend and guess our next term in the quotient:

enter image source here

and repeat for the last:

enter image source here

Which confirms that #x-3# is a factor of our polynomial evidenced by the fact that we get a zero remainder. So we conclude that:

#(2x^3-3x^2-5x-12) -: (x-3) = 2x^2+3x+4#