Binomial Theorem - One Term

Binomial Theorem - Find One Term - How it Works - Video

Binomial Theorem Expanded

Binomial Theorem Expanded:

Here we have expanded our binomial (a + b). The 1st line has 1 term. The 2nd line has 2 terms. The 3rd line has 3 terms. The 4th line has 4 terms, and so on.

If we take a look at the coefficients in front of each term, the second row has 1 then 1. The 3rd row has 1, 2, 1. The 4th row has 1, 3, 3, 1. Now, instead of distributing a bunch of times, we can use Pascal's triangle to to find the coefficients in any row. With the magic of math, Pascal's Triangle is our friend.

There is a pattern with the variables as well. Can you find one?

Properties of the Binomial Theorem

Properties of the Binomial Theorem:

The first property states what we stated already in another way. The power outside the parenthesis tells us how many terms we have after simplifying everything. In the case of (a + b)⁰. The 0 is n so n + 1 = 0 + 1 = 1 so there is 1 term. If we have (a + b)⁷. N is 7 so n + 1 = 7 + 1 = 8 so there are 8 terms.

The second property states that the first variable in the parenthesis starts with power outside the parenthesis and decreases each time, while the second variable does the opposite. Finally, when we add the two powers, the result will be same as a power at the beginning.

The third property states how it looks. There must be integers for our powers.

The forth property states another way on how to find the coefficient of the next term.

(k + 1)st Term Formula

(k +1)st Term Formula:

We can use the formula to find any term instead of distributing until we can no longer simplify.

Example 1A

Example 1A:

For line 1 - We have followed the pattern using the first and the second property to write out our first line.

For line 2 - We simplified by distributing the power to number or variable in the parenthesis.

For line 3 - We simplified the numbers with powers.

For line 4 - We combined like terms.

So our original power is 3, n + 1 = 4. We have 4 terms. We want the 3rd term. Our first term is 8x³ because it has the same power as original power. Then following property 2, our 2nd term is -60x². Our 3rd term is 150x, and our 4th term is -125.

So the final answer is 150x.

Example 1B

Example 1B:

We already know that is answer is 150x, but sometimes the power is something ridiculously high and it would take forever to distribute the binomial using the properties with the help of Pascal's triangle or not. We can use the (k + 1)st term formula.

We can use C(n, k) * a^n-k * b^k. So n = power = 3 and k = 3rd term = 3. We substitute and simplify and our result is 150x just like example 1A.

Example 2

Example 2:

We can use the (k + 1)st term formula instead of the binomial theorem to find our 8th term.

We can use C(n, k) * a^n-k * b^k. So n = power = 12 and k = 8th term = 8. We substitute and simplify and our result is 192,456x¹⁰ y^7/2 .

You might notice that we 7/2 for the power of y and that is not an integer, but that is okay because the results can be not an integer. However, k must be an integer, 8 in this case.

Example 3

Example 3:

We can use the (k + 1)st term formula instead of the binomial theorem to find the term involving n⁸.

Our first step is to find k. We know the term that we want has n⁸. So the question we should be asking ourselves is how we got there. Our original variable starts as n² power. We can set (n²)^k = n⁸ to find k so k is 4.

We can use C(n, k) * a^n-k * b^k to find our term. So n = power = 4 and k = 4th term = 4. We substitute and simplify and our result is (105/32)m⁶ n⁸ .

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