(a%5E3-3ab-b%5E2).jpeg "(a+b)(a^3-3ab-b^2)")
Expanding the Expression: (a+b)(a^3-3ab-b^2)
This expression involves multiplying two binomials: (a+b) and (a^3 - 3ab - b^2). We can achieve this using the distributive property or the FOIL method.
Distributive Property
The distributive property states that for any numbers a, b, and c: a(b+c) = ab + ac
Using this property, we can expand the expression as follows:
(a + b)(a^3 - 3ab - b^2) = a(a^3 - 3ab - b^2) + b(a^3 - 3ab - b^2)
Now, we distribute each term outside the parentheses:
= a^4 - 3a^2b - ab^2 + ba^3 - 3ab^2 - b^3
Simplifying the Expression
Finally, we combine like terms:
= a^4 + ba^3 - 3a^2b - 4ab^2 - b^3
Therefore, the expanded and simplified form of the expression (a+b)(a^3-3ab-b^2) is a^4 + ba^3 - 3a^2b - 4ab^2 - b^3.
Using the FOIL Method
The FOIL method is a shortcut for multiplying two binomials. It stands for First, Outer, Inner, Last. It follows these steps:
- First: Multiply the first terms of each binomial.
- Outer: Multiply the outer terms of the binomials.
- Inner: Multiply the inner terms of the binomials.
- Last: Multiply the last terms of each binomial.
Applying this method to our expression:
- First: a * a^3 = a^4
- Outer: a * -b^2 = -ab^2
- Inner: b * a^3 = ba^3
- Last: b * -b^2 = -b^3
Combining these terms, we get:
= a^4 - ab^2 + ba^3 - b^3
This result can be further simplified by combining like terms, as done in the previous method, giving us the same final answer: a^4 + ba^3 - 3a^2b - 4ab^2 - b^3.
Both methods yield the same result, so choose the method that you find easier to understand and apply.