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(1+2x+x2)4 can be expanded into a0+a1x+a2x2+…+a8x8. What is the value of a0+a1+a2+…+a8

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(1+2x+x2)4 can be expanded into a0+a1x+a2x2+…+a8x8. What is the value of a0+a1+a2+…+a8

by Max@Math Revolution » Wed Apr 29, 2020 4:28 am

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[GMAT math practice question]

(1+2x+x^2)^4 can be expanded into a0+a1x+a2x^2+…+a8x^8. What is the value of a0+a1+a2+…+a8?

A. 16
B. 32
C.64
D. 128
E. 256
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Source: — Problem Solving |
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Re: (1+2x+x2)4 can be expanded into a0+a1x+a2x2+…+a8x8. What is the value of a0+a1+a2+…+a8

by Max@Math Revolution » Fri May 01, 2020 2:48 am
=>

Since (1+2x+x^2)^4= a0+a1x+a2x^2+…+a8x^8, we get the value a0+a1+a2+…+a8 by substituting 1 for x and we have a0+a1x+a2x^2+…+a8x^8=(1+2·1+1^2)^4= 4^4=256.

Therefore, E is the answer.
Answer: E
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Re: (1+2x+x2)4 can be expanded into a0+a1x+a2x2+…+a8x8. What is the value of a0+a1+a2+…+a8

by deloitte247 » Fri May 01, 2020 6:48 am
Given that => $$\left(1+2x+x^2\right)^4=a_0+a_1\left(x\right)+a_2\left(x^2\right)+........+a_8\left(x^8\right)$$
$$If\ x\ =\ 1,\ then,\ a_0+a_1\left(x\right)+a_2\left(x^2\right)+........+a_8\left(x^8\right)$$
$$=>\ a_0+a_1\left(1\right)+a_2\left(1^2\right)+........+a_8\left(1^8\right)$$
$$=>\ a_0+a_1+a_2+........+a_8$$
$$hence,\ \left(1+2x+x^2\right)^4=a_0+a_1x+a_2x^2+........+a_8x^8$$
$$=a_0+a_1+a_2+........+a_8$$
$$Substituting(x=1)in\left(1+2x+x^2\right)^4$$
$$=\left(1+2x+x^2\right)^4$$
$$=\left[1+2\left(1\right)+\left(1^2\right)\right]^4$$
$$=\left(1+2+1\right)^4$$
$$=4^4=256$$
$$\sin ce\left(1+2x+x^2\right)^4=a_0+a_1+a_2+........+a_8$$
$$256=a_0+a_1+a_2+........+a_8$$
$$a_0+a_1+a_2+........+a_8=256$$

Answer = E
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