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Question 196091: Could someone maybe help me out with these two please:
How do I use synthetic division to divide x^4+4x^3+6x^2+4x+1 by x+1?
x^3-2x^2+x-3=0 what are the possible rational roots of this equation? (I dont really at all understand how to find out the rational roots here)
If possible could someone also maybe check if this problem is right: If x varies inversely as y and x= 2 when y=8, find x when y=17. x= 16/17? (since I sort of doubt that it is right, although I did it as my teacher showed on the board).
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
Let's deal with your last question first. I'd like to know what it is that makes you doubt that your answer is correct, especially since you apparently followed the process correctly. Yes, that is the correct answer.
Synthetic Division
First off, let's examine your problem:
I see coefficients of 1, 4, 6, 4, 1. Does that sequence look familiar to you? If not, look up Pascal's Triangle on Wikipedia (http://en.wikipedia.org/wiki/Pascal_triangle) and carefully examine the 4th row.
Your polynomial is the expansion of ^4) so when we divide by  we should expect to get ^3 = x^3 + 3x^2 + 3x + 1) . If you only needed the answer, then we would be done, but you need to see the synthetic division process...
Draw the lower left corner of a rectangle tall enough to contain two rows of numbers and long enough to contain the list of coefficients:
In regular polynomial long division, you would be dividing by a potential factor of your polynomial, in this case . But in synthetic division, you divide by a potential zero of the polynomial -- in this case -1 (because  ). So:
Bring down the first number in the dividend to below the division symbol:
Multiply the divisor times the number you just brought down and put the product under the second number in the dividend:
Add:
Multiply:
Add:
Multiply and add:
Multiply and add one more time:
Which is just what we expected, coefficients of 1, 3, 3, 1 and a remainder of 0.
Hence, the actual quotient is  .
Possible Rational Zeros
Rational Zero Theorem: If a polynomial function, written in descending order, has integer coefficients, then any rational zero must be of the form ± p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
So for
 or 
and
Hence the only possible rational zeros of the given polynomial equation are  or 
John
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