Difference between revisions of "1974 AHSME Problems/Problem 3"
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<math> \mathrm{(A)\ } -8 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 6 \qquad \mathrm{(D) \ } -12 \qquad \mathrm{(E) \ }\text{none of these} </math> | <math> \mathrm{(A)\ } -8 \qquad \mathrm{(B) \ }12 \qquad \mathrm{(C) \ } 6 \qquad \mathrm{(D) \ } -12 \qquad \mathrm{(E) \ }\text{none of these} </math> | ||
− | ==Solution== | + | ==Solution 1== |
Let's write out the multiplication, so that it becomes easier to see. | Let's write out the multiplication, so that it becomes easier to see. | ||
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We can now see that the only way to get an <math> x^7 </math> is by taking three <math> -x^2 </math> and one <math> 2x </math>. There are <math> \binom{4}{1}=4 </math> way to pick which term the <math> 2x </math> comes from, and the coefficient of each one is <math> (-1)^3(2)=-2 </math>. Therefore, the coefficient of <math> x^7 </math> is <math> (4)(-2)=-8, \boxed{\text{A}} </math>. | We can now see that the only way to get an <math> x^7 </math> is by taking three <math> -x^2 </math> and one <math> 2x </math>. There are <math> \binom{4}{1}=4 </math> way to pick which term the <math> 2x </math> comes from, and the coefficient of each one is <math> (-1)^3(2)=-2 </math>. Therefore, the coefficient of <math> x^7 </math> is <math> (4)(-2)=-8, \boxed{\text{A}} </math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | |||
+ | Note that <math>(1+2x-x^2)^4=((1-x)^2)^4 = (1-x)^8</math>. By the binomial theorem, the <math>x^7</math> term is <math>\binom{8}{7} (-1)^1 (x)^7 = -8x</math>. Therefore the coefficient is <math>\boxed{\textbf{-8}}</math>. | ||
==See Also== | ==See Also== |
Revision as of 21:29, 14 January 2014
Contents
[hide]Problem
The coefficient of in the polynomial expansion of
is
Solution 1
Let's write out the multiplication, so that it becomes easier to see.
We can now see that the only way to get an is by taking three and one . There are way to pick which term the comes from, and the coefficient of each one is . Therefore, the coefficient of is .
Solution 2
Note that . By the binomial theorem, the term is . Therefore the coefficient is .
See Also
1974 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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All AHSME Problems and Solutions |
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