# 5.8 – Binomial Theorem

Key Terms

• Binomial Coefficient – The coefficient of a given term in a binomial expansion.
• The coefficient of the k-term in the expansion of $(x+a)^n$ is denoted $\binom{n}{k}$.
• Binomial Expansion – An expression that is equivalent to a binomial raised to a positive integer power, and that results from carrying out the multiplication indicated by the power.
• Combinations – Unique selections of items, no repeats.
• Ex. If you have 3 different fruits: apple, pear, and banana, and must choose only 2, you wouldn’t choose [apple and pear], then [pear and apple] because those would be repeats.  The combination of apple and pear will only happen once, no matter the order.
• Choosing 2, you can also have a pear with banana or an apple with banana.
• Binomial Theorem – A polynomial identity which states: $(x+y)^n=\binom{n}{0}x^n+ \binom{n}{1}x^{n-1}y^1+ \binom{n}{2}x^{n-2}y^2+ \binom{n}{3}x^{n-3}y^3+...+ \binom{n}{n}y^n$
• Factorial – The product of n and all positive whole numbers less than n.
• The notation n! means “n factorial.”
• Ex. 4! = 4 3 2 1 = 24
• Ex. 8! = 8  4 3 2 1 = 40,320
• There is a button on your calculator “n!” you can use!
• Pascal’s Triangle – A triangular arrangement of numbers in which each row gives you the coefficients of a binomial expansion.
• Any number in the triangle can be computed by taking the sum of the two adjacent numbers in the previous row.
• Permutation – A selection of objects in which the order of the objects matters.
• Use this to show how many ways can you arrange a certain number of items.
• $\frac{n!}{k!(n-k)!}$
• 0! = 1

Notes

• Pascal’s Triangle
• Do you see a diagonal pattern?
• 1 + 1 = 2
• 1 + 2 = 3
• 1 + 3 = 4, so the coefficient of the 2nd term in the 4th line down will be 4.
• What will the 2nd term look like including the variables?  Look for the pattern. Try writing it out on paper!  Scroll ALL the way down to see the answer!
• Do you see a horizontal pattern (for the exponents)?
• From left to right, powers of x get smaller while powers of y get bigger
• The row will always start and end with the same power (which matches the row that it is in)
• The pattern found with Pascal’s triangle works for any binomial
• Using Pascal’s triangle may be faster than using the FOIL method when binomials have coefficients.

• Watch and copy the video below

• Ex. $(2+y)^3$
• Look at the 3rd row of Pascal’s Triangle and replace ALL x’s with 2’s.
• Substitute 2 for x:  $1(2)^3+3(2)^2y+3(2)y^2+1y^3$
• Simplified: $8+12y+6y^2+y^3$

• Ex. $(3x-2y)^2$
• Rewrite as: $(3x+(-2y))^2$
• Substitute 3x for x and substitute -2y for y: $1(3x)^2+2(3x)(-2y)+(-2y)^2$
• Simplify: $9x^2-12xy+4y^2$

• Pascal’s Patterns Enumerated
1. Expanding a binomial to the nth power gives n + 1 terms.
2. The first term of $(x+y)^n$ will always be $x^n$, and the last will always be $y^n$.
3. The sum of the exponents of each term of the expansion of $(x+y)^n$ is n.
4. The exponents of the x-terms in the expansion of $(x+y)^n$ will start at n and decrease by 1 for every consecutive term.
5. The exponents of the y-terms in the expansion of $(x+y)^n$ will start at zero and increase by 1 for every consecutive term.
6. The coefficients follow the pattern of Pascal’s triangle for the nth row.

• Negative Values for Pascal’s Triangle Expansion
• For $(x-y)^n$, raising a negative value to a power creates a pattern of alternating signs.
• For $((-x)+(-y))^n$, if n is an even number, every sign will be positive; but, if n is an odd number, every sign will be negative.

• Ex. $(x-y)^n$

• Pascal’s Triangle by Row (n) and Column (k)
• Rows: x
• Columns: y

• Permutations Using the Combination Formula
• $\binom{n}{k}$ is read “n choose k,”
• Formula can be used to find the kth term of the nth row of Pascal’s triangle
• It is defined by the quotient (ratio) of factorials
• $\binom{n}{k}=\frac{n!}{k!(n-k)!}$
• And for some reason, 0! = 1
• Use a calculator to find the answers

• Ex. You can choose 3 out of 26 friends to go with you to the movies.  How many possible combinations can you have?
• Since calculators can’t handle numbers that large without converting to scientific notation, here’s a shortcut!

Examples

• For $(x+y)^6$
• Ex 1a. Which of the following could NOT be a term in the expansion?
• $6x^5y$
• $15x^2y^4$
• $20xy$
• $15x^3y^3$
• Answer: since each term’s exponents MUST add up to 6, “20xy” is the one that could NOT be a term in the expansion.
• Ex 1b. What is the expansion?

• Ex 2. Calculate the 5th term for $(x-3y)^8$
• The first term will have 8 as the x-exponent and zero as the y-exponent.
• The fifth term will have 8−4 (so = 4) as the x-exponent and 4 as the y-exponent.
• The term will be positive because $(-3y)^4$ is positive.
• The coefficient from the triangle is 70, so the fifth term is:
• $70x^4(-3y)^4=(70)(81)x^4y^4=5670x^4y^4$

• Pascal’s Triangle: The Next Level

• Coefficients of Pascal’s Triangle