Simplify the following expressions into simplest terms:
(x^3 + 4x^2y + 4xy^2 + 2xy - 6y^2) / (x + 2y)
To simplify the given expression, we can use the method of polynomial long division.
Step 1: Write the dividend (the expression we want to simplify) and the divisor.
Dividend: x^3 + 4x^2y + 4xy^2 + 2xy - 6y^2
Divisor: x + 2y
Step 2: Perform long division.
Start by dividing the first term of the dividend (x^3) by the first term of the divisor (x). This gives us x^2, which we write on top.
x^2
Multiply the divisor (x + 2y) by x^2, and subtract it from the dividend:
x^3 + 4x^2y + 4xy^2 + 2xy - 6y^2
- (x^3 + 2x^2y)
Simplifying, we get:
2x^2y + 4xy^2 + 2xy - 6y^2
Step 3: Repeat the process.
Bring down the next term from the dividend (-2x^2y) and divide it by the first term of the divisor (x). This gives us - 2xy, which we write underneath the first term on top.
x^2 - 2xy
Multiply the divisor (x + 2y) by -2xy, and subtract it from the previous result:
2x^2y + 4xy^2 + 2xy - 6y^2
- (2x^2y - 4xy^2)
Simplifying, we get:
6xy^2 + 2xy - 6y^2
Step 4: Repeat the process.
Bring down the next term from the dividend (6xy^2) and divide it by the first term of the divisor (x). This gives us 6y^2, which we write underneath the previous result on top.
x^2 - 2xy + 6y^2
Multiply the divisor (x + 2y) by 6y^2, and subtract it from the previous result:
6xy^2 + 2xy - 6y^2
- (6xy^2 + 12y^3)
Simplifying, we get:
2xy - 12y^3
Step 5: Repeat the process.
Bring down the next term from the dividend (2xy) and divide it by the first term of the divisor (x). This gives us 2y, which we write underneath the previous result on top.
x^2 - 2xy + 6y^2 + 2y
Multiply the divisor (x + 2y) by 2y, and subtract it from the previous result:
2xy - 12y^3
- (2xy + 4y^2)
Simplifying, we get:
-4y^2 - 12y^3
Step 6: There are no more terms in the dividend to bring down. We have our simplified expression:
x^2 - 2xy + 6y^2 + 2y - 4y^2 - 12y^3
Combining like terms, we can simplify it further:
x^2 - 2xy + 2y - 4y^2 + 6y^2 - 12y^3
Finally, we have:
x^2 - 2xy + 2y - 4y^2 + 6y^2 - 12y^3
Therefore, the simplified expression is x^2 - 2xy + 2y - 4y^2 + 6y^2 - 12y^3.