How do you multiply #(2a + 2a^2)(3 + a)#?

1 Answer
Apr 6, 2018

Answer:

#2a^3 + 8a^2 + 6a#

Explanation:

To multiply this polynomial, you must use the distributive property. Recall that a polynomial like #4(x + 2) = 4(x) + 4(2) = 4x + 8#.

To use the distributive property in a polynomial like the one you gave, it helps to "simplify" it to an easier form. Let's let #u = (3 + a)#, that way we have less to keep track of. Then we have:

#(2a + 2a^2)(3+a) = (2a + 2a^2)(u) = u(2a + 2a^2)#.

Now we can use the familiar distributive property:

#u(2a + 2a^2) = u(2a) + u(2a^2) = 2au + 2a^2u#.

We now have #u# in our answer, which we don't want. Remember that we let #u = 3 + a#, so we can replace every #u# with a #3 + a#.

This gives:

#2au + 2a^2u = 2a(3+a) + 2a^2(3+a)#.

We can see that we now have to use the distributive property again, twice this time. This gives:

#2a(3+a) + 2a^2(3+a) = (6a + 2a^2) + (6a^2 + 2a^3)#,
#= 2a^3 + 6a^2 + 2a^2 + 6a = 2a^3 + 8a^2 + 6a#.

Thus, our final answer is #2a^3 + 8a^2 + 6a#.