Certainly! Let's verify the equality [tex]\( x^3 + y^3 = (x + y)(x^2 - xy + y^2) \)[/tex] step-by-step.
To verify this, we’ll simplify both sides of the equation separately and see if they are equal.
1. Simplify the left-hand side (LHS):
The left-hand side is already in its simplest form:
[tex]\[ LHS = x^3 + y^3 \][/tex]
2. Simplify the right-hand side (RHS):
The right-hand side is given by:
[tex]\[ RHS = (x + y)(x^2 - xy + y^2) \][/tex]
We need to expand this product. We'll do this step-by-step:
First, apply the distributive property to multiply [tex]\( (x + y) \)[/tex] by each term inside the parentheses [tex]\( (x^2 - xy + y^2) \)[/tex].
[tex]\[ (x + y)(x^2 - xy + y^2) = x(x^2 - xy + y^2) + y(x^2 - xy + y^2) \][/tex]
Let's expand this:
[tex]\[ x(x^2 - xy + y^2) = x^3 - x^2y + xy^2 \][/tex]
[tex]\[ y(x^2 - xy + y^2) = yx^2 - xy^2 + y^3 \][/tex]
Now, add the two results together:
[tex]\[ x^3 - x^2y + xy^2 + yx^2 - xy^2 + y^3 \][/tex]
Next, combine like terms:
[tex]\[ x^3 + y^3 - x^2 y + x^2 y + xy^2 - xy^2 \][/tex]
Notice that [tex]\( -x^2 y \)[/tex] and [tex]\( x^2 y \)[/tex] cancel each other out, and [tex]\( xy^2 \)[/tex] and [tex]\( -xy^2 \)[/tex] also cancel each other:
[tex]\[ x^3 + y^3 \][/tex]
3. Comparison:
- Left-hand side: [tex]\( x^3 + y^3 \)[/tex]
- Right-hand side: [tex]\( x^3 + y^3 \)[/tex]
Since both sides simplify to the same expression, we conclude that:
[tex]\[ x^3 + y^3 = (x + y)(x^2 - xy + y^2) \][/tex]
Thus, the equality is verified!
