To expand and simplify the expression [tex]\( y = (x+2)^3 - 3 \)[/tex], follow these steps:
1. Expand the binomial expression:
The first part of the expression is [tex]\((x + 2)^3\)[/tex]. We can use the binomial theorem to expand this:
[tex]\[
(x + 2)^3 = x^3 + 3x^2(2) + 3x(2^2) + 2^3
\][/tex]
2. Calculate each term:
- [tex]\( x^3 \)[/tex] stays as [tex]\( x^3 \)[/tex].
- [tex]\( 3x^2 \cdot 2 = 6x^2 \)[/tex].
- [tex]\( 3x \cdot 2^2 = 3x \cdot 4 = 12x \)[/tex].
- [tex]\( 2^3 = 8 \)[/tex].
Combining all these, we get:
[tex]\[
(x + 2)^3 = x^3 + 6x^2 + 12x + 8
\][/tex]
3. Subtract 3 from the expanded expression:
Now, we need to subtract 3 from this expanded form:
[tex]\[
x^3 + 6x^2 + 12x + 8 - 3
\][/tex]
4. Combine constants:
Combine the constants [tex]\( 8 \)[/tex] and [tex]\( -3 \)[/tex]:
[tex]\[
8 - 3 = 5
\][/tex]
So, the final expanded form of [tex]\( y \)[/tex] is:
[tex]\[
y = x^3 + 6x^2 + 12x + 5
\][/tex]
Thus, the expanded form of the given function [tex]\( y = (x + 2)^3 - 3 \)[/tex] is:
[tex]\[
y = x^3 + 6x^2 + 12x + 5
\][/tex]
