Section 4.2 Expanding Binomials Using Pascal’s Triangle © Copyright all rights reserved to Homework depot: www.BCMath.ca

Blaise pascale and Pascal’s triangle Blaise Pascal (June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher He used the triangle to solve problems in probability theory. Numbers used in Pascal’s triangle was discovered by many mathematicians before him but with different applications The earliest explicit depictions of the triangle occur in the 10th century in commentaries on the Chandas Shastra (India) In Iran, it is known as "Khayyam triangle” (1048-1131); , finding nth roots of a binomial expansion (section 6.6) In China, it is known as “Yang Hui's triangle"(1238-1298) In Italy, they call it "Tartaglia's triangle", used in solving cubic polynomials(1500-1577)

Pascal’s triangle 1 Begin with three 1’s on the top 1 1 Each number is the sum of the values directly above 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 1 10 45 120 210 252 210 120 45 10 1

Expanding Binomials Expand the following Binomials: The coefficients of all the terms corresponds to the numbers in Pascal’s Triangle! © Copyright all rights reserved to Homework depot: www.BCMath.ca

Binomial Theorem: When expanding a binomial in the form of (a+b)n, we can use Pascal’s triangle to determine the coefficients of each term Rules: The exponent “n” to determines which row to use from Pascal’s triangle  use the “n+1” row There will be “n+1” terms in the expansion The first term must have a power of an, with the “a” variable descending in degree by one In addition, the first term will have b0, with each term ascending in degree by one Note: for every nth power, there will be n+1 terms © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Expand the following binomial: n = 4, so use the 5th row in Pascal’s triangle The second variable “b” will be ascending in power, starting with 0 The first variable “a” will be descending in power, starting with 4 n = 5, so use the 6th row in Pascal’s triangle The second variable “b” will be ascending in power, starting with 0 The first variable “a” will be descending in power, starting with 4 Simplify each term!

Practice: Expand the following Binomial i) Indicate how many terms there are ii) what is the coefficient of the term with x3 n = 6, so there will be 7 terms in the expansion Use the 7th row in Pascal’s Triangle Coefficient of the term with x3 is 20 n = 3, so there will be 4 terms in the expansion Coefficient of the term with x3 is 1 Use the 3rd row in Pascal’s Triangle Simplify the exponents © Copyright all rights reserved to Homework depot: www.BCMath.ca

Ex: Find the constant term: “the constant term” has no variables Expand the binomial The constant term is 24

Ex: Solve for “a” and “b”

Binomial Thm with Decimal Expansions One advanced application of the binomial theorem is finding the decimal expansion of numbers with large exponents: Ex: What are the last two digits in the decimal expansion of 1350 ? This is a going to be a big number, question is asking for the last two digits from the right of this number The only term that we need to be concerned with is the last one We don’t need to worry about these terms b/c they will all have 2 zeroes at the far right Last two digits is 49

Which of the following is bigger? Justify your solution

Amc 12 2006 asa