Sure! Let's solve the expression step-by-step:
Given the expression:
[tex]\[
(x - y) \left(10x^2 + 25y^2 - 35xy - x + y\right)
\][/tex]
We need to expand this expression by distributing [tex]\((x - y)\)[/tex] through the polynomial inside the parentheses. Let's do this one term at a time.
1. Multiply [tex]\(x\)[/tex] by each term in the polynomial:
[tex]\[
x \cdot (10x^2 + 25y^2 - 35xy - x + y)
\][/tex]
[tex]\[
= 10x^3 + 25xy^2 - 35x^2y - x^2 + xy
\][/tex]
2. Multiply [tex]\(-y\)[/tex] by each term in the polynomial:
[tex]\[
-y \cdot (10x^2 + 25y^2 - 35xy - x + y)
\][/tex]
[tex]\[
= -10x^2y - 25y^3 + 35xy^2 + yx - y^2
\][/tex]
3. Combine all the terms together:
[tex]\[
10x^3 + 25xy^2 - 35x^2y - x^2 + xy - 10x^2y - 25y^3 + 35xy^2 + yx - y^2
\][/tex]
4. Group and combine like terms:
[tex]\[
10x^3
\][/tex]
[tex]\[
(-35x^2y - 10x^2y - x^2) = -45x^2y - x^2
\][/tex]
[tex]\[
(25xy^2 + 35xy^2 + xy + yx) = 60xy^2 + 2xy
\][/tex]
[tex]\[
-25y^3
\][/tex]
[tex]\[
-y^2
\][/tex]
Putting it all together, we get the expanded expression:
[tex]\[
10x^3 - 45x^2y - x^2 + 60xy^2 + 2xy - 25y^3 - y^2
\][/tex]
This is the fully expanded form of the given expression.
