Synthetic Division

[0313phd]

Problem: 9x to the 23rd power, minus 7x to the 10th power, minus 2x to the fifth power, plus 1 IS DIVIDED BY X +1. What is the remainder?
I know you can solve this through synthetic division, but you must leave zeros as placeholders. There is such a range between the exponents that I end up filling an entire page turned width wise, and it takes forever. I still get the answer wrong because I think I miss something in my impatience. Is there a quicker or less messy way of solving this either through synthetic division, regular division, or whatever? If I am just impatient, I will work on that. 0313

 

[JeffM]

Problem: 9x to the 23rd power, minus 7x to the 10th power, minus 2x to the fifth power, plus 1 IS DIVIDED BY X +1. What is the remainder?

Does the problem statement ask for the quotient or the remainder? In either case, let me ask you a question. Consider a sum of n summands divided by a divisor. Does it make any difference whether you first divide the individual summands by the divisor and then add up the n resulting quotients or first add up the n summands and divide that sum by the divisor?

I know you can solve this through synthetic division, but you must leave zeros as placeholders. There is such a range between the exponents that I end up filling an entire page turned width wise, and it takes forever. I still get the answer wrong because I think I miss something in my impatience. Is there a quicker or less messy way of solving this either through synthetic division, regular division, or whatever? If I am just impatient, I will work on that. 0313

 

[Mrspi]

Problem: 9x to the 23rd power, minus 7x to the 10th power, minus 2x to the fifth power, plus 1 IS DIVIDED BY X +1. What is the remainder?
I know you can solve this through synthetic division, but you must leave zeros as placeholders. There is such a range between the exponents that I end up filling an entire page turned width wise, and it takes forever. I still get the answer wrong because I think I miss something in my impatience. Is there a quicker or less messy way of solving this either through synthetic division, regular division, or whatever? If I am just impatient, I will work on that. 0313

As I read your question, your question is what is the remainder?

If that's correct, than the EASIEST way to find the remainder when a polynomial P(x) is divided by (x - r) is to use the remainder theorem:

If a polynomial P(x) is divided by (x - r), the remainder is a CONSTANT, P(r). That is, substitute r for each x in the polynomial, and evaluate. The value you get will be the remainder.

Example: What is the remainder when x[sup:3l9flk78]3[/sup:3l9flk78] - 4x[sup:3l9flk78]2[/sup:3l9flk78] - 7x + 10 is divided by (x + 5)? We need that divisor in the form x - r. (x + 5) is the same thing as x - (-5). So, to find the remainder, evaluate the polynomial when x=-5:

(-5)[sup:3l9flk78]3[/sup:3l9flk78] - 4(-5)[sup:3l9flk78]2[/sup:3l9flk78] - 7(-5) + 10
-125 - 100 + 35 + 10
-225 + 45
-180

The remainder, then, when you divide x[sup:3l9flk78]3[/sup:3l9flk78] - 4x[sup:3l9flk78]2[/sup:3l9flk78] - 7x + 10 by (x + 5) should be -180. You can verify that it IS -180 by doing the division.

Try the same process on your division...remember that (x + 1) is the same thing as x - (-1), so you'll want to evaluate your polynomial with -1 substituted for x.

 

[mmm4444bot]

9x to the 23rd power, minus 7x to the 10th power, minus 2x to the fifth power, plus 1

:idea: This is not a good way to express a polynomial because it reads like so:

(9x)^23 - (7x)^10 - (2x)^5 + 1 which is actually the polynomial 8862938119652501095929x^23 - 282475249x^10 - 32*x^5 + 1

What you were trying to state in English (above) is:

"9 times x to the 23rd power, minus 7 times x to the 10th power, minus 2 times x to the fifth power, plus 1"

YET, we previously showed you how to state powers using the caret symbol (^), and lookagain already explained to you this same issue in another thread.

(9x)^23 is not the same as 9x^23.

Please type polynomials using math instead of English:

9x^23 - 7x^10 - 2x^5 + 1




Is there a quicker or less messy way of solving this either through synthetic division (No), regular division (No), or whatever? (Yes - Mrs. Pi explained how)

 

[0313phd]

THANK YOU SO MUCH for the help. 0313