Sure, let's simplify the expression [tex]\((2x - y)^3\)[/tex] step by step. We can use the binomial theorem to expand the cubic polynomial. The binomial theorem states that:
[tex]\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\][/tex]
In this case, our values are:
[tex]\[ a = 2x \][/tex]
[tex]\[ b = y \][/tex]
We need to plug these values into the binomial expansion formula. Let's go through each term one by one.
1. First term: [tex]\(a^3\)[/tex]
[tex]\[
(2x)^3 = 8x^3
\][/tex]
2. Second term: [tex]\(- 3a^2b\)[/tex]
[tex]\[
-3 \cdot (2x)^2 \cdot y = -3 \cdot 4x^2 \cdot y = -12x^2y
\][/tex]
3. Third term: [tex]\(3ab^2\)[/tex]
[tex]\[
3 \cdot (2x) \cdot y^2 = 6xy^2
\][/tex]
4. Fourth term: [tex]\(- b^3\)[/tex]
[tex]\[
-y^3
\][/tex]
Now, combining all these terms, we get:
[tex]\[
(2x - y)^3 = 8x^3 - 12x^2y + 6xy^2 - y^3
\][/tex]
So, the simplified form of the expression [tex]\((2x - y)^3\)[/tex] is:
[tex]\[
8x^3 - 12x^2y + 6xy^2 - y^3
\][/tex]
