I have been working on a problem "Find the sum of all the coefficients of the expansion of $P(x)=(x−2)^{100}$." Knowing that the polynomial is far too large to calculate, I tried to find a way to find each coefficient individually, but so far, I have been unsuccessful. Does anyone know what steps I have to take to calculate the sum of the coefficients?
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7If you plug $x=1$ into $ax^2+bx+c$, what do you get? – B. Goddard Jun 29 '19 at 19:01
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Knowing that $$(x-2)^{100}=\sum_{k=0}^{100}\binom{100}k(-2)^kx^{100-k}$$ we see that substituting $x=1$ will yield the sum of the coefficients, which is $(-1)^{100}=1$.
Parcly Taxel
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Hint: If the polynomial were written as $$p(x)=a_nx^n+\cdots+a_0,$$ how would you determine the sum of its coefficients?
Clayton
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