Difference between revisions of "1961 AHSME Problems/Problem 5"
(Created page with 'Let <math>S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1</math>. Then S equals: (A) <math>(x-2)^4</math> (B) <math>(x-1)^4</math> (C) <math>x^4</math> (D) <math>(x+1)^4</math> (E) <math>x…') |
Rockmanex3 (talk | contribs) (Solution to Problem 5) |
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| − | + | == Problem == | |
| − | + | Let <math>S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1</math>. Then <math>S</math> equals: | |
| − | (B) | + | |
| − | + | <math>\textbf{(A)}\ (x-2)^4 \qquad | |
| − | + | \textbf{(B)}\ (x-1)^4 \qquad | |
| − | + | \textbf{(C)}\ x^4 \qquad | |
| + | \textbf{(D)}\ (x+1)^4 \qquad | ||
| + | \textbf{(E)}\ x^4+1 </math> | ||
| + | |||
| + | ==Solution== | ||
| + | |||
| + | Let <math>y = x-1</math>. Substitution results in | ||
| + | <cmath>S = y^4 + 4y^3 + 6y^2 + 4y + 1</cmath> | ||
| + | <cmath>S = (y+1)^4</cmath> | ||
| + | Substituting back results in | ||
| + | <cmath>S = x^4</cmath> | ||
| + | The answer is <math>\boxed{\textbf{(C)}}</math>. This problem can also be solved by traditionally expanding and combining like terms (though it would take much longer). | ||
| + | |||
| + | ==See Also== | ||
| + | {{AHSME 40p box|year=1961|num-b=4|num-a=6}} | ||
| + | |||
| + | {{MAA Notice}} | ||
| + | |||
| + | [[Category:Introductory Algebra Problems]] | ||
Latest revision as of 18:54, 17 May 2018
Problem
Let 
. Then 
equals:
Solution
Let 
. Substitution results in
![\[S = y^4 + 4y^3 + 6y^2 + 4y + 1\]](http://latex.artofproblemsolving.com/4/3/8/43815d981fba2bc62ed1649641f8538a8ea1b86c.png)
![\[S = (y+1)^4\]](http://latex.artofproblemsolving.com/3/a/6/3a68e57d0b438c214700d43cfce82cee6ecf8010.png)
Substituting back results in
![\[S = x^4\]](http://latex.artofproblemsolving.com/e/a/4/ea4b6399335f691d78233ab83492a507caefbafe.png)
The answer is 
. This problem can also be solved by traditionally expanding and combining like terms (though it would take much longer).
See Also
| 1961 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
