SOLUTION: What is the remainder when the polynomial {{{9x^23-7x^12-2x^5+1}}} divided by x+1

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: What is the remainder when the polynomial {{{9x^23-7x^12-2x^5+1}}} divided by x+1      Log On


   

Question 377441: What is the remainder when the polynomial 9x%5E23-7x%5E12-2x%5E5%2B1 divided by x+1
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The straightforward way to find a remainder is to divide. But because of the range of exponents in the polynomial this is not the simple way to find the remainder. After all, look at what the synthetic division would look like:
-1 | 9 0 0 0 0 0 0 0 0 0 0 -7 0 0 0 0 0 0 -2 0 0 0 0 1
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This will work but it is tedious.

The fast way to find the remainder is to take advantage of the Remainder Theorem and advantage of how easy it is to raise -1 to high powers. The Remainder Theorem tells us that if one divides a polynomial by (x-r) then the remainder is the value of that polynomial for x = r. So the remainder when 9x%5E23-7x%5E12-2x%5E5%2B1 is divided by (x+1), or (x-(-1)), is the same as the value of 9x%5E23-7x%5E12-2x%5E5%2B1 when x = -1. All we need to do then is to evaluate 9x%5E23-7x%5E12-2x%5E5%2B1 for x = -1:
9%28-1%29%5E23-7%28-1%29%5E12-2%28-1%29%5E5%2B1
(Note if x was not 1 or -1, this method is not easy because raising other numbers to the 23rd, 12th and 5th powers are not simple.)
Simplifying...
9(-1) - 7(+1) - 2(-1) + 1
-9 - 7 + 2 + 1
-13
The value of the polynomial, when x = -1, is -13. So the remainder when dividing the polynomial by (x+1) will also be -13.