Expert: Scotto - 10/28/2009

Question
Hello. I would really appreciate the quick help! and explanations as well so I can understand it as much as possible.    
1. Use long division to divide x^3+7x^2+6x-18 by x+3    

2. For f(x) = 2x^3+ x^2- 7x- 6 use the rational zero test to find the possible real zeros. Test the values to determine the actual real zeroes.   

3. How many possible rational roots are there for the function f(x)=3x^3+bx^2-cx-6, where b and c are integers?    

4. Solve the equation  3x^3 - 26x^2 + 61x – 30 = 0 given that 3 is a zero of the function f(x)=x^3-26x^2    
+61x – 30    
{3,-5,-2/3}    
{3,5,2/3}    
{3,2,5/3}    
{3,-2,-5/3}     

Thank you so much!
~AL

Answer
1. Use long division to divide x^3+7x^2+6x-18 by x+3
Take 1 3 and divide it into 1 7 6 –18 be checking the first number in both strings.
1/1 = 1, and 1(1 3) = 1 3, so subtract off (1 3) from (0 4 6 –18).
This gives (4 6 –18) when the leading 0 is dropped.

4/1=4, and 4(1 3) = 4 12.  (4 6 –18) – (4 12) = (4-4, 6-12 –19) = (0 –6 –18).
Again, drop the zero and get (-6 –18).

(1 3) * -6 = (-6 –18).
So the answer is (x+3)(x²+4x-6).


2. For f(x) = 2x^3+ x^2- 7x- 6 use the rational zero test to find the possible real zeros. Test the values to determine the actual real zeroes.
Put in x=1.  The answer is 2+1-7-6=-10.
Try x=-1.  The answer is –2+1+7-6=8-8=0.
This means we can divide by (x+1).
The result so far would be (x+1)(2x²-x-6).
Factoring the last term gives us (x+1)(2x+3)(x-2).


3. How many possible rational roots are there for the function f(x)=3x^3+bx^2-cx-6, where b and c are integers?
There are 3 rational roots since the highest power is 3.


4. Solve the equation 3x^3 - 26x^2 + 61x – 30 = 0
given that 3 is a zero of the function f(x)=x^3-26x^2 +61x – 30

3 is not zero of that function.
If you put in 3 you get 27 - 26•9 +61•3 – 30 = 27 – 234 + 183 – 30 =210 – 264 = -54.

I'm not sure what {3,-5,-2/3}, {3,5,2/3}, {3,2,5/3}, and {3,-2,-5/3} are here for.
Maybe those are suppose to be the roots.  Was there a typo in the problem?