Binomial Theorem

Algebra Level 3

( x 2 − 1 − 3 x ) 18 + ( 37 − 32 x ) 2 − ( 2 x 99 + 3 15 ) + ( 2 x 3 − 3 x ) 14 (x^2-1-3x)^{18}+(37-32x)^2-(2x^{99}+3^{15})+(2x^3-3x)^{14}

If the sum of all the coefficients of the polynomial above is C C , then find C − 2 C-2 .

25 ( 3 10 − 2 ) − 14 25(3^{10}-2)-14 0 0 25 − C 2 25-C^2 25 25 3 15 − 2 3^{15}-2 3 15 3^{15} 3 14 − 2 3^{14}-2 25 ( 3 15 + 1 ) + 3 15 − 3 25(3^{15}+1)+3^{15}-3

Anandmay Patel by Anandmay Patel , 20, India

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3 solutions

Jesse Nieminen
Oct 9, 2016

We use the fact that for any polynomial p ( x ) p\left(x\right) , the sum of the coefficients is equal to p ( 1 ) p(1) . (Exercise: Prove this fact)

Let's now define P ( x ) = ( x 2 − 1 − 3 x ) 18 + ( 37 − 32 x ) 2 − ( 2 x 99 + 3 15 ) + ( 2 x 3 − 3 x ) 14 P\left(x\right) = (x^2-1-3x)^{18}+(37-32x)^2-(2x^{99}+3^{15})+(2x^3-3x)^{14}

P ( 1 ) = ( 1 2 − 1 − 3 ⋅ 1 ) 18 + ( 37 − 32 ⋅ 1 ) 2 − ( 2 ⋅ 1 99 + 3 15 ) + ( 2 ⋅ 1 3 − 3 ⋅ 1 ) 14 = 3 18 + 5 2 − 2 − 3 15 + 1 = 27 ⋅ 3 15 − 3 15 + 25 − 3 + 2 = 25 ( 3 15 + 1 ) + 3 15 − 3 + 2 = C \begin{aligned} P\left(1\right) &= (1^2-1-3\cdot1)^{18}+(37-32\cdot1)^2-(2\cdot1^{99}+3^{15})+(2\cdot1^3-3\cdot1)^{14} \\ &=3^{18}+5^2-2-3^{15}+1 \\ &=27\cdot3^{15} - 3^{15} + 25 - 3 + 2 \\ &=25\left(3^{15} + 1\right) + 3^{15} - 3 + 2 \\ &=C \end{aligned}

Hence, C − 2 = 25 ( 3 15 + 1 ) + 3 15 − 3 C - 2 = 25\left(3^{15} + 1\right) + 3^{15} - 3 , and since other choices clearly are smaller, this one must be the only correct answer.

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The complete solution.

Anandmay Patel - 4 years, 8 months ago
Prince Loomba
Oct 9, 2016

Put x=1, the sum of all the terms is equal to the sum of coefficients as no coefficient is altered. Answer is clear.

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Anandmay Patel
Oct 9, 2016

HINT: \textbf{HINT:} Rather than expanding the expression through binomial theorem,just put \text{Rather than expanding the expression through binomial theorem,just put} x = 1 x=1 in the expression \text{in the expression} . Wondering why?Think! \text{Wondering why?Think!}

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Our answers combined give the solution!

Prince Loomba - 4 years, 8 months ago

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