Certainly! Let's expand the given algebraic expression: [tex]\((2x^2 - x - 1)^3\)[/tex].
To expand this, we apply the binomial theorem for a cubic expansion, which involves distributing and combining terms in a structured manner. Instead of calculating each step manually, I’ll explain the process that leads us to the expanded form:
1. Start with the expression:
[tex]\[(2x^2 - x - 1)^3\][/tex]
2. Cube the expression:
Expanding [tex]\((2x^2 - x - 1)^3\)[/tex] involves finding the product of [tex]\((2x^2 - x - 1)\)[/tex] with itself three times:
[tex]\[(2x^2 - x - 1)(2x^2 - x - 1)(2x^2 - x - 1)\][/tex]
3. First, expand [tex]\((2x^2 - x - 1)(2x^2 - x - 1)\)[/tex]:
We multiply each term in the first polynomial by each term in the second polynomial:
[tex]\[
(2x^2 - x - 1)(2x^2 - x - 1) = (2x^2)(2x^2) + (2x^2)(-x) + (2x^2)(-1) + (-x)(2x^2) + (-x)(-x) + (-x)(-1) + (-1)(2x^2) + (-1)(-x) + (-1)(-1)
\][/tex]
Simplifying this:
[tex]\[
= 4x^4 - 2x^3 - 2x^2 - 2x^3 + x^2 + x - 2x^2 + x + 1
\][/tex]
Combine like terms:
[tex]\[
= 4x^4 - 4x^3 - 3x^2 + 2x + 1
\][/tex]
4. Next, multiply the result by the remaining [tex]\((2x^2 - x - 1)\)[/tex]:
We then take our result [tex]\(4x^4 - 4x^3 - 3x^2 + 2x + 1\)[/tex] and multiply it by [tex]\(2x^2 - x - 1\)[/tex]:
[tex]\[
(4x^4 - 4x^3 - 3x^2 + 2x + 1)(2x^2 - x - 1)
\][/tex]
The expansion involves distributing each term of the first polynomial to each term of the second polynomial. After carefully combining like terms, the simplified result is:
[tex]\[
8x^6 - 12x^5 - 6x^4 + 11x^3 + 3x^2 - 3x - 1
\][/tex]
Therefore, the expanded form of [tex]\((2x^2 - x - 1)^3\)[/tex] is:
[tex]\[
\boxed{8x^6 - 12x^5 - 6x^4 + 11x^3 + 3x^2 - 3x - 1}
\][/tex]
