Certainly! Let's expand and simplify the expression [tex]\((2x^4 - y^5)^2\)[/tex].
1. Identify the expression to be expanded:
[tex]\[
(2x^4 - y^5)^2
\][/tex]
2. Recall the algebraic identity for the square of a binomial:
[tex]\[
(a - b)^2 = a^2 - 2ab + b^2
\][/tex]
For our expression, [tex]\(a = 2x^4\)[/tex] and [tex]\(b = y^5\)[/tex].
3. Substitute [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the identity:
[tex]\[
(2x^4 - y^5)^2 = (2x^4)^2 - 2(2x^4)(y^5) + (y^5)^2
\][/tex]
4. Calculate each term individually:
- First term:
[tex]\[
(2x^4)^2 = 4x^8
\][/tex]
- Second term:
[tex]\[
-2(2x^4)(y^5) = -4x^4y^5
\][/tex]
- Third term:
[tex]\[
(y^5)^2 = y^{10}
\][/tex]
5. Combine all terms to write the expanded expression:
[tex]\[
4x^8 - 4x^4y^5 + y^{10}
\][/tex]
6. State the expanded form in standard polynomial form:
[tex]\[
4x^8 - 4x^4y^5 + y^{10}
\][/tex]
Therefore, the expanded polynomial is:
[tex]\[
4x^8 - 4x^4y^5 + y^{10}
\][/tex]
