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(a+b)(a^3-3ab-b^2)

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2 Answers

Hey Dwight. This kind of thing looks messy, but it becomes much more simple when you break it down.
Multiplying two polynomials means multiplying each term in the first expression by each term in the second expression, and then adding them all together. Let's take it just a little bit at a time.
First we will multiply A by the second expression, but we'll do this one term at a time.
A * A3 = A4 Remember that multiplying similar variables is the same as adding exponents. A1 + A3 = A4 because 1 + 3 = 4
A * -3AB = -3A2B Multiplying different variables is the same as multiplying a variable and a constant. 5 * x is as simple as that expression can get, so it just stays at 5x (we take out the multiplication symbol to make it easier to read. It's still there though).
A * -b2 = -AB2 This one is the same thing as the second one. Remember that one negative in an expression will make the entire expression negative, which is why we can move the - symbol to the front.

Now we need to add these all together. Using the same rules, we come up with A4 - 3A2B - AB2
 
Now we are going to do the same thing with the second part of our first expression. We've already done something similar, so this will be really easy.
B * A3 = A3B
B * -3AB = -3AB2
B * -B2 = -B3
 
Adding those together gives us A3B - 3AB2 - 3B3
And we are going to add this to the earlier part of our answer, for a complete answer of
A4 + A3B - 3A2B - 4AB2 - B3
1) In order to multiply two polynomials, one must multiply each term in the second polynomial by each term in the first one and add the results, also combining the like terms. In your case, first polynomial has two terms: a and b. Second polynomial has three terms: a3, -3ab, and -b2. Now let us multiply each of those three by the first term of the first polynomial, that is, a.
 
a*a3=a4
a*(-3ab)=-3a2b
a*(-b2)=-ab2
 
Next, do the same with the second term of the first polynomial, that is b.
b*a3=ba3
b*(-3ab)=-3ab2
b*(-b2)=-b3
 
Now we shall add all six results together:
(a+b)(a3-3ab-b2)=a4-3a2b-ab2+ba3-3ab2-b3;
Now we collect like terms. Those are terms that has the same variable structure. In your case those are the third and the fifth terms in the product, -ab2 and -3ab2.  Notice, they both have a to the first power, and b to the second power, multiplied together. Their sum is -4ab2. Notice also, that the second term, though looking similar, is not the same, since it has a raised to the second power and b raised to just the first power. Therefore, it cannot be combined with the other two we just added together. So, we finally obtain:
 
(a+b)(a3-3ab-b2)=a4-3a2b-4ab2+ba3-b3;

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