SOLUTION: Use the Binomial Theorem to expand each binomial and express the result in simplified form. (3x + y)^3

Question 283682: Use the Binomial Theorem to expand each binomial and express the result in simplified form.
(3x + y)^3
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

Start with the given expression

To expand this, we're going to use binomial expansion. So let's look at Pascal's triangle:
1
1 1
1 2 1
1 3 3 1

Looking at the row that starts with 1,3, etc, we can see that this row has the numbers:

1, 3, 3, and 1

These numbers will be the coefficients of our expansion. So to expand , simply follow this procedure:
Write the first coefficient. Multiply that coefficient with the first binomial term and then the second binomial term . Repeat this until all of the coefficients have been written.

Once that has been done, add up the terms like this:

Notice how the coefficients are in front of each term.

However, we're not done yet.

Looking at the first term , raise to the 3rd power and raise to the 0th power.

Looking at the second term raise to the 2nd power and raise to the 1st power.

Continue this until you reach the final term.

Notice how the exponents of are stepping down and the exponents of are stepping up.

So the fully expanded expression should now look like this:

Distribute the exponents

Multiply

Multiply the terms with their coefficients

So expands and simplifies to .

In other words,
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