How do you square a binomial?

Let’s use as a general binomial, and square it:

Next let's show that this pattern will work for all types of binomials:

There are a few things to notice about the pattern:

• If there is a constant or coefficient in either term, it is squared along with the variables.



• The powers variable in the first term of the binomial descend in an orderly fashion.

2nd degree, 1st degree, 0 degree or 4th degree, 2nd degree, 0 degree

• The powers of the variable in the second term ascend in an orderly fashion.

0 degree, 1st degree, 2nd degree

• The sign of the 2nd term is negative in the 3rd example, as it should be.



• The sum of the exponents for every term in the expansion is 2.



• There are 3 terms in the 2nd power expansion.

What if we cube a binomial?

There are a few things to notice about the pattern:

• If there is a constant or coefficient in either term, it is raised to the appropriate power along with the variables.



• The powers of the variable in the first term of the binomial descend in an orderly fashion.

3rd degree, 2nd degree, 1st degree, 0 degree

• powers of the variable in the second term ascend in an orderly fashion.

0 degree, 1st degree, 2nd degree, 3rd degree

• The signs of the 2nd and 4th term are appropriately negative in the 2nd example.



• The sum of the exponents in each term of the expansion are 3.



• There are 4 terms in the 3rd degree expansion.

Summarizing: What patterns do we need to do any binomial expansion?

• The powers of the first term (the “a” term) descend in consecutive order , starting with the power of the expansion and ending with the zero power . Note that we raise the entire term to that power, then one lower, etc.



• The powers of the second term (the “b” term) ascend in consecutive integer order, starting with zero power and ending with the power of the expansion.



• The sum of the exponents (before simplifying them) of each term is the same as the power of the expansion



• You will always have one more term than the number of the expansion.



• The signs of an will alternate positive, then negative, etc.
• The pattern of the coefficients follows Pascal’s Triangle:

Let's Practice:

1. Expand

Start with the first term

Note there is no need to show the because it is 1

The 2nd term will be

Note that the exponents add up to 5,

The 3rd term will be

The 4th term will be

The 5th term will be

The last term will be

Again there is no need to show the because it is 1.

Therefore

2. Expand

The degree is 5 so we will have six terms altogether. The coefficients needed to complete the expansion are the 1 5 10 10 5 1 row of Pascal’s Triangle.

Start with the first term

As usual, there is no need to show the because it is 1.

The 2nd term will be

Note that the exponents add up to 5.

The 3rd term will be

The 4th term will be

The 5th term will be

The last term will be

All together we get: