Difference between revisions of "2011 PuMAC Problems/Algebra Problem A1"
(→Solution) |
|||
| Line 4: | Line 4: | ||
Find the sum of the coefficients of the polynomial <math>(63x-61)^4</math>. | Find the sum of the coefficients of the polynomial <math>(63x-61)^4</math>. | ||
| − | == Solution == | + | == Solution 1 == |
Note that in a polynomial <math>\sum^{n}_{i=0} a_ix^i</math>, <math>x=1</math> gives <math>\sum^{n}_{i=0} a_i</math>, which is the sum of the coefficients. | Note that in a polynomial <math>\sum^{n}_{i=0} a_ix^i</math>, <math>x=1</math> gives <math>\sum^{n}_{i=0} a_i</math>, which is the sum of the coefficients. | ||
Therefore, the sum of the coefficients of <math>(63x-61)^4</math> is <math>(63-61)^4 = 2^4 = \boxed{16}</math> | Therefore, the sum of the coefficients of <math>(63x-61)^4</math> is <math>(63-61)^4 = 2^4 = \boxed{16}</math> | ||
| + | == Solution 2 == | ||
| + | Expanding, we get <math>15752961x^4 - 61011468x^3 + 88611894x^2 - 57199212x + 13845841</math>. Summing the coefficients, we have <math>15752961 - 61011468 + 88611894 - 57199212 + 13845841 = \boxed{16}</math> | ||
Latest revision as of 20:15, 6 August 2023
Contents
Problem
Find the sum of the coefficients of the polynomial 
.
Solution 1
Note that in a polynomial 
, 
gives 
, which is the sum of the coefficients.
Therefore, the sum of the coefficients of is 
Solution 2
Expanding, we get 
. Summing the coefficients, we have 
