(x%5E2%2B4y%5E2%2B2xy%2B6y-3x%2B9).jpeg "(x-2y+3)(x^2+4y^2+2xy+6y-3x+9)")
Expanding the Expression (x-2y+3)(x^2+4y^2+2xy+6y-3x+9)
This article explores the expansion of the given expression: (x-2y+3)(x^2+4y^2+2xy+6y-3x+9). We will use the distributive property and perform the necessary multiplications to arrive at the simplified polynomial.
Applying the Distributive Property
The distributive property states that for any three expressions a, b, and c: a(b + c) = ab + ac. We can apply this property to our expression by multiplying each term within the first set of parentheses by each term within the second set.
Let's break down the multiplication process:
Step 1: Multiply x by each term in the second set:
- x(x^2) = x^3
- x(4y^2) = 4xy^2
- x(2xy) = 2x^2y
- x(6y) = 6xy
- x(-3x) = -3x^2
- x(9) = 9x
Step 2: Multiply -2y by each term in the second set:
- -2y(x^2) = -2x^2y
- -2y(4y^2) = -8y^3
- -2y(2xy) = -4xy^2
- -2y(6y) = -12y^2
- -2y(-3x) = 6xy
- -2y(9) = -18y
Step 3: Multiply 3 by each term in the second set:
- 3(x^2) = 3x^2
- 3(4y^2) = 12y^2
- 3(2xy) = 6xy
- 3(6y) = 18y
- 3(-3x) = -9x
- 3(9) = 27
Combining Like Terms
Now that we have all the individual products, we need to combine the like terms.
After combining like terms, the expanded expression becomes:
x^3 + 4xy^2 - 8y^3 + 9x + 12y^2 - 18y + 27
Therefore, the expanded form of the expression (x-2y+3)(x^2+4y^2+2xy+6y-3x+9) is x^3 + 4xy^2 - 8y^3 + 9x + 12y^2 - 18y + 27.