Question

Simply the following expressions:
1. X - [2x- (3x- (4x- 5x))]
2. ( 2x + 1)(x + 1)squared
3. ( x + 3 )cubed

Answer 1

Let's simplify each expression one by one.

1. **Expression:** \( X - [2x - (3x - (4x - 5x))] \)

First, simplify the innermost parentheses:
\[
4x - 5x = -x
\]
Now we have:
\[
3x - (-x) = 3x + x = 4x
\]
Next, we simplify \( 2x - 4x \):
\[
2x - 4x = -2x
\]
Finally, substitute back into the expression:
\[
X - (-2x) = X + 2x
\]
So, the simplified expression is:
\[
X + 2x
\]

2. **Expression:** \( (2x + 1)(x + 1)^2 \)

First, simplify \( (x + 1)^2 \):
\[
(x + 1)(x + 1) = x^2 + 2x + 1
\]
Then, multiply that result by \( (2x + 1) \):
\[
(2x + 1)(x^2 + 2x + 1)
\]
Now distribute \( (2x + 1) \):
\[
= 2x(x^2 + 2x + 1) + 1(x^2 + 2x + 1)
\]
Expanding each part:
\[
= 2x^3 + 4x^2 + 2x + x^2 + 2x + 1
\]
Combine like terms:
\[
= 2x^3 + 5x^2 + 4x + 1
\]
So, the simplified expression is:
\[
2x^3 + 5x^2 + 4x + 1
\]

3. **Expression:** \( (x + 3)^3 \)

We can expand this using the binomial theorem or by recognizing it as \( (a + b)^3 \):
\[
= a^3 + 3a^2b + 3ab^2 + b^3
\]
Here, \( a = x \) and \( b = 3 \):
\[
= x^3 + 3(x^2)(3) + 3(x)(3^2) + 3^3
\]
Calculating each term:
\[
= x^3 + 9x^2 + 27x + 27
\]
So, the simplified expression is:
\[
x^3 + 9x^2 + 27x + 27
\]

In summary, the simplified expressions are:
1. \( X + 2x \)
2. \( 2x^3 + 5x^2 + 4x + 1 \)
3. \( x^3 + 9x^2 + 27x + 27 \)