how to get from step 2 to step 3
1. (x^3 + y^3) + xy(x + y)
2. (x + y)(x^2 - xy + y^2) + xy(x + y)
3. (x + y)(x^2 + y^2)
objective: factorization
2. (x + y)(x^2 - xy + y^2) + xy(x + y)
2. a)
= (x + y)(x^2 - xy + y^2 + xy)
x+y was a common factor ...
3. (x + y)(x^2 + y^2)
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To understand how to get from step 2 to step 3 in the factorization process of the expression (x^3 + y^3) + xy(x + y), let's break down the steps involved:
Step 1: Recognize that the given expression consists of two terms, (x^3 + y^3) and xy(x + y).
Step 2: Identify any common factors that can be factored out. In this case, we can factor out the common factor of (x + y) from both terms.
Step 3: Apply the distributive property. Distribute the common factor (x + y) to the terms within the original expression.
In equation form, the factorization would look like this:
(x^3 + y^3) + xy(x + y)
= (x + y)(x^2 - xy + y^2) + xy(x + y)
Now, let's move from step 2 to step 3:
Step 4: Apply the distributive property to the first term, (x + y)(x^2 - xy + y^2), by distributing (x + y) within the brackets.
(x + y)(x^2 - xy + y^2) + xy(x + y)
= x(x^2 - xy + y^2) + y(x^2 - xy + y^2) + xy(x + y)
Step 5: Combine like terms. In this case, we can combine the terms that have the same exponents:
x(x^2) + x(-xy) + x(y^2) + y(x^2) + y(-xy) + y(y^2) + xy(x) + xy(y)
= x^3 - x^2y + xy^2 + x^2y - xy^2 + y^3 + x^2y + y^2x
= x^3 + y^3 + y^2x + x^2y + xy^2
Step 6: Rearrange the terms to group like terms together.
x^3 + y^3 + y^2x + x^2y + xy^2
= (x^3 + y^3) + (x^2y + y^2x + xy^2)
Step 7: Apply the commutative property of addition to reorder the terms within the parentheses.
(x^3 + y^3) + (x^2y + y^2x + xy^2)
= (x^3 + y^3) + (xy^2 + x^2y + y^2x)
Step 8: Observe that within the second set of parentheses, we have the same terms as the original expression: xy(x + y).
(x^3 + y^3) + (xy^2 + x^2y + y^2x)
= (x^3 + y^3) + xy(x + y)
Step 9: Finally, substitute back into the original expression:
(x^3 + y^3) + xy(x + y)
= (x^3 + y^3) + xy(x + y)
= (x + y)(x^2 - xy + y^2) + xy(x + y)
Therefore, we have successfully obtained step 3: (x + y)(x^2 + y^2), which is the factorized form of the given expression.