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Welcome to our Math lesson on How to Expand Binomials in Higher Powers, this is the second lesson of our suite of math lessons covering the topic of Binomial Expansion and Coefficients, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Expanding Binomials in Higher Powers
We can still use the FOIL Rule (which gives the two above expansions) to find how to expand higher power binomials. For example, we can expand a binomial in the fourth power in the following way:
(a + b)4 = (a + b)2 ∙ (a + b)2
= (a2 + 2ab + b2 ) ∙ (a2 + 2ab + b2 )
= a2 a2 + a2 ∙ 2ab + a2 ∙ b2 + 2ab ∙ a2 + 2ab ∙ 2ab + 2ab ∙ b2 + b2 ∙ a2 + b2 ∙ 2ab + b2 b2
= a4 + 2a3 b + a2 b2 + 2a3 b + 4a2 b2 + 2ab3 + a2 b2 + 2ab3 + b4
= a4 + (2a3 b + 2a3 b) + (a2 b2 + 4a2 b2 + a2 b2 ) + (2ab3 + 2ab3 ) + b4
= a4 + 4a3 b + 6a2 b2 + 4ab3 + b4
The expansion of a binomial, when raised to the fifth power, is even more challenging, as
(a + b)5 = (a + b)3 ∙ (a + b)2
= (a3 + 3a2 b + 3ab2 + b3 ) ∙ (a2 + 2ab + b2 )
= a3 ∙ a2 + a3 ∙ 2ab + a3 ∙ b2 + 3a2 b ∙ a2 + 3a2 b ∙ 2ab + 3a2 b ∙ b2 + 3ab2 ∙ a2 + 3ab2 ∙ 2ab + 3ab2 ∙ b2 + b3 ∙ a2 + b3 ∙ 2ab + b3 ∙ b2
= a5 + 2a4 b + a3 b2 + 3a4 b + 6a3 b2 + 3a2 b3 + 3a3 b2 + 6a2 b3 + 3ab4 + a2 b3 + 2ab4 + b5
= a5 + (2a4 b + 3a4 b) + (a3 b2 + 6a3 b2 + 3a3 b2 ) + (3a2 b3 + 6a2 b3 + a2 b3 ) + (3ab4 + 2ab4 ) + b5
= a5 + 5a4 b + 10a3 b2 + 10a2 b3 + 5ab4 + b5
As you see, the more the power of a binomial increases, the longer the expansion procedure becomes. Therefore, it is necessary to find an alternative method of expanding binomials raised to a given power. In fact, the problem is not the value of powers, as you may see from the above examples that the power of the first term decreases by 1 when moving from left to right while the power of the second term increases by 1 during the same procedure. The main challenge to overcome in this respect consists of the value of the coefficients preceding each term after the expansion. In other words, it is already clear that the general form of a binomial raised to the nth power is
(a + b)n = c1 ∙ an ∙ b0 + c2 ∙ an - 1 ∙ b1 + c3 ∙ an - 2 ∙ b2 + ⋯ + cn - 1 ∙ a1 ∙ bn - 1 + cn ∙ a0 ∙ bn
where c1, c2, c3, …, cn - 1 and cn are the coefficients preceding the corresponding terms of the given binomial after the expansion.
From the examples discussed earlier, it is obvious that c1 and cn are always 1 regardless of the power of the binomial.
The first scientist who found a solution to this issue was Blaise Pascal, who proposed his famous triangle (Pascal's Triangle) for finding the binomial coefficients, which we will explain in the following part of this tutorial.
More Binomial Expansion and Coefficients Lessons and Learning Resources
Sequences and Series Learning Material| Tutorial ID | Math Tutorial Title | Tutorial | Video
Tutorial | Revision
Notes | Revision
Questions |
|---|
| 12.3 | Binomial Expansion and Coefficients | | | | |
| Lesson ID | Math Lesson Title | Lesson | Video
Lesson |
|---|
| 12.3.1 | The Square and the Cube of a Binomial | | |
| 12.3.2 | How to Expand Binomials in Higher Powers | | |
| 12.3.3 | Using Pascal's Triangle | | |
| 12.3.4 | Working with Binomial Coefficients and the Limitation of Pascal's Triangle | | |
| 12.3.5 | The Binomial Coefficients Theorem | | |
| 12.3.6 | What is a Binomial when it is not in the Standard Form | | |
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