Expand Everything?

Algebra Level 3

x ( x ( x ( x ( x ( x ( x ( x + 2 ) + 3 ) + 4 ) + 5 ) + 6 ) + 7 ) + 8 ) + 9 x(x(x(x(x(x(x(x+2) +3 )+ 4)+5) + 6) + 7) + 8)+9

What is the coefficient of x 4 x^4 in the expansion above?


The answer is 5.

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1 solution

A way to determine the coefficient of x 4 x^4 is by expanding the innermost algebraic expression, which is x ( x + 2 ) + 3 = x 2 + 2 x + 3 x(x+2)+3 = x^2+2x+3 Note that there are 6 6 preceding x x 's of the expression above. From the previous statement, we are now certain that the highest degree of the polynomial is 8 8 . It now implies that its expansion would start with x 6 ( x 2 + 2 x + 3 ) = x 8 + 2 x 7 + 3 x 6 x^6(x^2+2x+3) = x^8+2x^7+3x^6 and so on, until reaching 9 9 . Therefore, the succeeding terms decrease in its degree. Hence by counting, we arrive at an answer of 5 \boxed{5} .

Moderator note:

Good observation of how to interpret the algebraic expression.

Very neat solution. Thank you! ;)

Pi Han Goh - 4 years, 11 months ago

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