Question

Distribute (a^2+2ab+b^2)(a+b)

Answer 1

To distribute the expression (a^2+2ab+b^2)(a+b), apply the distributive property to multiply each term of the trinomial by each term of the binomial, resulting in a^3 + 3a^2b + 3ab^2 + b^3.

Step-by-step explanation:

The student is asking to distribute the expression (a^2+2ab+b^2)(a+b). To perform this distribution, you apply the distributive property, which in general terms can be stated as A(B + C) = AB + AC. For vectors and complex numbers, this property ensures that you can distribute multiplication over addition or subtraction.

In this specific case, you have a polynomial multiplied by a binomial. You need to multiply each term of the polynomial by each term of the binomial, and then combine like terms if necessary. The expression (a^2+2ab+b^2) is a perfect square trinomial, and it will be distributed across (a+b).

Here's the step-by-step distribution:

1. Multiply a^2 by a: a^3.
2. Multiply a^2 by b: a^2b.
3. Multiply 2ab by a: 2a^2b.
4. Multiply 2ab by b: 2ab^2.
5. Multiply b^2 by a: ab^2.
6. Multiply b^2 by b: b^3.

Now, combine the like terms:

• a^3 (no like terms to combine).
• a^2b + 2a^2b: These are like terms, so you can combine them to get 3a^2b.
• 2ab^2 + ab^2: These are like terms as well, so you can combine them to get 3ab^2.
• b^3 (no like terms to combine).

The final distributed answer is: a^3 + 3a^2b + 3ab^2 + b^3