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Algebra and Applications Test 3

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Question : 16 of 25

Marks: +1, -0

If

x = 2 - 2^{1/3} + 2^{2/3}

, then the value of

x^3 - 6x^2 + 18x + 18

is:

22
33
40
45

Solution:

Concept:

Use substitution to simplify the cube-root expression and find the required polynomial value.

Explanation:

Let

a=2^{1/3}

.

Then

x=2-a+a^2

, so

x-2=a^2-a

.

Now cube

x-2

:

(a^2-a)^3=(a^2)^3-a^3-3(a^2)(a)(a^2-a).

Since

a^3=2

, this becomes

(x-2)^3=4-2-6(a^2-a)=2-6(x-2).

Expand the left side:

x^3-6x^2+12x-8=2-6x+12.

Simplify:

x^3-6x^2+18x-22=0.

So,

x^3-6x^2+18x=22.

Therefore,

x^3-6x^2+18x+18=22+18=40.

Answer:

\boxed{40}

(Option C)

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