(a-b)(a%2B2b).jpeg "(2a-b)(a-b)(a+2b)")
Expanding the Expression (2a-b)(a-b)(a+2b)
This article will walk you through the process of expanding the expression (2a-b)(a-b)(a+2b). We'll use the distributive property and other algebraic techniques to arrive at the simplified form of the expression.
Step 1: Expanding the first two factors
First, we need to expand the product of the first two factors: (2a-b)(a-b).
We can use the FOIL method (First, Outer, Inner, Last) to accomplish this:
- First: 2a * a = 2a²
- Outer: 2a * -b = -2ab
- Inner: -b * a = -ab
- Last: -b * -b = b²
Combining these terms gives us:
(2a-b)(a-b) = 2a² - 2ab - ab + b²
Simplifying:
(2a-b)(a-b) = 2a² - 3ab + b²
Step 2: Expanding the entire expression
Now we have to multiply the result from step 1 by the remaining factor (a+2b):
(2a² - 3ab + b²)(a+2b)
We can use the distributive property to multiply each term in the first expression by each term in the second:
- 2a² * a = 2a³
- 2a² * 2b = 4a²b
- -3ab * a = -3a²b
- -3ab * 2b = -6ab²
- b² * a = ab²
- b² * 2b = 2b³
Combining these terms gives us:
(2a² - 3ab + b²)(a+2b) = 2a³ + 4a²b - 3a²b - 6ab² + ab² + 2b³
Step 3: Simplifying the expression
Finally, we can combine the like terms to get the simplified form of the expression:
**(2a² - 3ab + b²)(a+2b) = 2a³ + a²b - 5ab² + 2b³
Therefore, the expanded form of the expression (2a-b)(a-b)(a+2b) is 2a³ + a²b - 5ab² + 2b³.