PreCalculus Section 8-5 The Binomial Theorem

Objectives Use the Binomial Theorem to calculate binomial coefficients. Use binomial coefficients to write binomial expansions. Use Pascal’s Triangle to calculate binomial coefficients.

Binomial Coefficients

We know that a binomial is a polynomial that has two terms. In this section, you will study a formula that provides a quick method of raising a binomial to a power. To begin, look at the expansion of (x + y) n.

The Binomial Expansion The binomial expansions of (x + y) n reveal several patterns.

Binomial Coefficients 1.The sum of the powers of each term is n. For instance, in the expansion of (x + y) 5 the sum of the powers of each term is 5. (x + y) 5 = x 5 + 5x 4 y x 3 y x 2 y 3 + 5x 1 y 4 + y = = 5 2.The coefficients increase and then decrease in a symmetric pattern. The coefficients of a binomial expansion are called binomial coefficients. 3.The expansion of (x + y) n begins with x n and ends with y n. 4.The variables in the terms after x n follow the pattern x n-1 y, x n-2 y 2, x n-3 y 3 and so on to y n. With each term the exponent on x decreases by 1 and the exponent on y increases by 1.

Binomial Coefficients To find them, you can use the Binomial Theorem.

! ! ! ! The Factorial Symbol 0! = 1 1! = 1 n! = n(n-1) ·... · 3 · 2 · 1 n must be an integer greater than or equal to 2 What this says is if you have a positive integer followed by the factorial symbol you multiply the integer by each integer less than it until you get down to 1. 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720 Your calculator can compute factorials. The ! symbol is under the “catalog" key.

Your Turn: Evaluate (a) 5! (b) 7! Check using your calculator. Solution (a) (b)

Binomial Coefficients Binomial Coefficient For nonnegative integers n and r, with r < n, The symbols and for the binomial coefficients are read “n choose r” Your calculator can compute these as well. It is also under the “catalog" key and n C r.

Example: Evaluating Binomial Coefficients Evaluate (a) (b) Solution (a) (b)

Your Turn: Finding Binomial Coefficients Find each binomial coefficient. Check using your calculator.

BINOMIAL EXPANSION

Expanding Binomials The Binomial Theorem The x's start out to the nth power and decrease by 1 in power each term. The y 's start out to the 0 power and increase by 1 in power each term. The binomial coefficients are found by computing the combinations. And the sum of the powers for x and y is n.

Example: Expanding Binomials Expand: (x + 2) x 5 + x 4 (2) 1 + x 3 (2) 2 + x 2 (2) 3 + x(2) 4 + (2) 5 = = x 5 + 5(x 4 )(2) + 10(x 3 )(4) + 10(x 2 )(8) + 5(x)(16) + 32 = x x x x x + 32

Your Turn: Write the expansion of the expression (x + 1) 3. Solution: Therefore, the expansion is as follows. The binomial coefficients are. (x + 1) 3 = x 3 + (3)x 2 (1) + (3)x(1 2 ) + (1 3 ) = x 3 + 3x 2 + 3x + 1

Your Turn: Write the binomial expansion of. Solution:

Expand: (3x – 2y) (3x) 4 + (3x) 3 (-2y) 1 + (3x) 2 (-2y) 2 + (3x) 1 (-2y) 3 + (-2y) 4 = = (81x 4 ) + 4(27x 3 )(-2y) + 6(9x 2 )(4y 2 ) + 4(3x)(-8y 3 ) + (16y 4 ) = 81x 4 – 216x 3 y + 216x 2 y 2 – 96xy y 4 Example:

Your Turn: Expand. Solution

Find the first three terms of the following expansion. (x 2 – 1) 20 = x (x 38 )(-1) + 190(x 36 )(1) +... = x 40 – 20x x 36 –... Your Turn:

Binomial Expansions

General term in the expansion is; n C r x n-r y r Or The Binomial Theorem –General Term Which is the (r + 1) th term

Example: Find the sixth term of (a + 2b) 8. Solution: Because the formula is for the (r + 1) th term, r is one less than the number of the term you need. So, to find the sixth term in this binomial expansion, use r = 5, n = 8, x = a and y = 2b. n C r x n – r y r = 8 C 5 a 8 – 5 (2b) 5 = 56  a 3  (2b) 5 = 56(2 5 )a 3 b 5 = 1792a 3 b 5

Your Turn: Find the fourth term of. Solution Using n = 10, r = 3, x = a, y = 2b in the formula, we find the fourth term is

Example: Find the 8 th term of the expansion of (2x – 3) (2x) 5 (-3) 7 8 th term = = 792(32x 5 )(-2187) = 1,647,360 x 7 y 8

Your Turn: In the expansion of (x + 2y) 15, find the term containing y 8. = 6435(x 7 )(256y 8 ) = 1,647,360 x 7 y 8

Pascal’s Triangle

Pascal “Contradiction is not a sign of falsity, nor the lack of contradiction a sign of truth.” Quoted in W H Auden and L Kronenberger, The Viking Book of Aphorisms (New York 1966).  June 19, 1623-August 19,1662  Born in Clermont-Ferrand, France  Mathematician, physicist, religious philosopher  Instrumental in development of economics and social science

History Pascal’s Triangle has a long line of history behind it. It might be understood that because it’s named after the mathematician Blaise Pascal, that he was the one that invented it. This however is not entirely true. There were many people that had worked with and even created a Pascal’s Triangle long before Pascal was even thought of. Don’t believe me? Click on the picture and learn more about the history of Pascal’s Triangle……. Blaise Pascal Or click here to see some more history…..here

Pascal’s Triangle There is a convenient way to remember the pattern for binomial coefficients. By arranging the coefficients in a triangular pattern, you obtain the following array, which is called Pascal’s Triangle. This triangle is named after the famous French mathematician Blaise Pascal (1623–1662) = = 21

Pascal’s Triangle The first and last number in each row of Pascal’s Triangle is 1. Every other number in each row is formed by adding the two numbers immediately above the number. Pascal noticed that the numbers in this triangle are precisely the same numbers as the coefficients of binomial expansions, as follows.

Pascal’s Triangle The top row of Pascal’s Triangle is called the zeroth row because it corresponds to the binomial expansion (x + y) 0 = 1. Similarly, the next row is called the first row because it corresponds to the binomial expansion (x + y) 1 = 1(x) + 1(y). In general, the nth row of Pascal’s Triangle gives the coefficients of (x + y) n.

Example: Using Pascal’s Triangle Use the seventh row of Pascal’s Triangle to find the binomial coefficients. 8 C 0 8 C 1 8 C 2 8 C 3 8 C 4 8 C 5 8 C 6 8 C 7 8 C 8 Solution:

Binomial Expansion using Pascal’s Triangle Expand (x+y) n There are three parts to consider (a)the coefficients from Pascal’s Triangle. (b)the powers of x (decreasing from n in above expansion to 0). (c)the powers of y (increasing from 0 in the above expansion). All elements in the expansion are in the form x n–r y r i.e. powers always add up to n.

Example: Give the expansion of (x + 1) 6 (a) Pascal’s Triangle We want the row starting 1 6 ……

Continued: Give the expansion of (x + 1) 6 (b) Powers of x (c) Powers of y So the full expansion is: (x + 1) 6 = x 6, x 5, x 4, ……. 1 0, 1 1, 1 2, … (ALL 1) 16x +15x 2 +20x 3 +15x 4 +6x 5 + 1x 6 +

Your Turn: Give the expansion of (1 - 3x) 5 (a) Pascal’s Triangle (b) Powers of “x” (c) Powers of “y” (Increasing) So the full expansion of (1 - 3x) All 1 1, (-3x), (-3x) 2, (-3x) 3, (-3x) 4, (-3x) 5 = 1+ 5 ∙(- 3x) + 10 ∙ (-3x) ∙ (-3x) ∙ (-3x) 4 + (-3x) 5 = 1- 15x + 90x x x x 5

Pascal's Triangle will give us the coefficients for a binomial expansion of any power if we extended it far enough. This is good for lower powers but could get very large. For larger powers it is easier to use combinations to calculate the binomial coefficients. Binomial Expansion with Pascal’s Triangle

Binomial Theorem Summary

Coefficients are arranged in a Pascal triangle. Summation of the indices of each term is equal to the power (order) of the expansion. The first term of the expansion is arranged in descending order after the expansion. The second term of the expansion is arranged in ascending order order after the expansion. Number of terms in the expansion is equal to the power of the expansion plus one. The Binomial Theorem Points to be remmebered :

The Binomial Theorem The General Term

Homework Section 8.5 pg 624 – 626; –Vocab. Check: #1 – 4 all –Exercises: #1 – 43 odd, 49 – 71 odd, 77