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In the expansion in powers of x of the function

\(\displaystyle (1+x)(a-bx)^{12}\)

the coefficient of the x-cubed term is zero. Find in its simplest form the value of the ratio a/b.

(It's there if you double click.)

 Jul 16, 2023
edited by Guest  Jul 16, 2023
 #1
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b*tch, im double clicking and i aint sein sh&t

 Jul 16, 2023
 #2
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? Works for me.

 Jul 16, 2023
 #3
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1 + x in brackets, a - bx in brackets, a - bx raised to the power 12.

 Jul 16, 2023
 #4
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a/b = 4/3.

 Jul 17, 2023
 #5
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By me, GA!

Guest Jul 17, 2023
 #7
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For the record, Post # 5, signed by GA, is not by the real GA.

Guest Jul 23, 2023
 #6
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\(\displaystyle (1+x)(a-bx)^{12} = \\ \displaystyle (1+x)\{(^{12}_{\, 0})a^{12}+( ^{12}_{\, 1})a^{11}(-bx)+( ^{12}_{ \,2})a^{10}(-bx)^{2}+( ^{12}_{ \, 3})a^{9}(-bx)^{3}+\dots \} \\ \displaystyle = (1+x)(a^{12}-12a^{11}bx+66a^{10}b^{2}x^{2}-220a^{9}b^{3}x^{3}+ \dots) \\ \displaystyle = a^{12}+(a^{12}-12a^{11}b)x-(12a^{11}b-66a^{10}b^{2})x^{2}+(66a^{10}b^{2}-220a^{9}b^{3})x^{3}+\dots \)

 

If the coefficient of the x-cubed term is zero, then

 

\(\displaystyle 66a^{10}b^{2}=220a^{9}b^{3} \\ \displaystyle \frac{a}{b}=\frac{220}{66}=\frac{10}{3}.\)

 Jul 18, 2023

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