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QuestionSimplify the expression: (2k+1)3−(1−2k)3−4k(4k2+3)(2k+1)^3 - (1-2k)^3 - 4k(4k^2+3).

Studdy Solution

STEP 1

Assumptions1. The expression is (k+1)3−(1−k)3−4k(4k+3)(k+1)^3 - (1-k)^3 -4k(4k^+3). We need to simplify this expression

STEP 2

First, let's expand the cubes in the expression. The formula for the cube of a binomial (a+b)^ is a^ +a^2b +ab^2 + b^ and for (a-b)^ is a^ -a^2b +ab^2 - b^.
(2k+1)^ = (2k)^ +(2k)^2(1) +(2k)(1)^2 + (1)^ (1-2k)^ = (1)^ -(1)^2(2k) +(1)(2k)^2 - (2k)^

STEP 3

Now, substitute the expanded forms back into the original expression.
(2k+1)3−(1−2k)3−k(k2+3)=(8k3+12k2+6k+1)−(1−6k+12k2−8k3)−k(k2+3) (2k+1)^3 - (1-2k)^3 -k(k^2+3) = (8k^3 +12k^2 +6k +1) - (1 -6k +12k^2 -8k^3) -k(k^2+3)

STEP 4

implify the expression by combining like terms.
=8k3+12k2+6k+1−1+6k−12k2+8k3−4k(4k2+3) =8k^3 +12k^2 +6k +1 -1 +6k -12k^2 +8k^3 -4k(4k^2+3)

STEP 5

implify further by combining like terms.
=16k3+12k−4k(4k2+3) =16k^3 +12k -4k(4k^2+3)

STEP 6

Expand the last term in the expression.
=16k3+12k−16k3−12k =16k^3 +12k -16k^3 -12k

STEP 7

implify the expression by combining like terms.
=16k3−16k3+12k−12k =16k^3 -16k^3 +12k -12k

STEP 8

implify further by combining like terms.
=0 =0 So, the simplified form of the given expression is0.

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