Expansion of Binomials

(x+y) n The expansion of a binomial follows a predictable pattern Learn the pattern and you can expand any binomial

What are we doing? Expanding binomials of the form (x+y) n Looking for patterns in the expansion of binomials Developing a method for expanding binomials

Why are we doing this? Topic in Intermediate and College Algebra Necessary in Calculus if not for the Chain Rule

What have we learned before that will help? Distributive property of real numbers Multiplying polynomials

How will I know if I have learned this? You will be able to expand any binomial of the form (x+y) n without the laborious task of successive multiplications of (x+y)

(x+y) 1 (x+y) 1 =x+y What is the degree of the expansion? How many terms are in the expansion? What is the exponent of x in the first term? What is the exponent of y in the first term? What is the sum of the exponents in the first term?

(x+y) 1 (x+y) 1 =x+y What is the exponent of x in the second term? What is the exponent of y in the second term? What is the sum of the exponents in the second term? What is the coefficient of the first term? What is the coefficient of the second term?

(x+y) 2 (x+y)(x+y)

(x+y) 2 =(x+y)(x+y) x+yx+y Write the first expression twice for the two terms in the second expression

(x+y) 2 =(x+y)(x+y) x+yx+y xxyy Place each term of the second expression below

(x+y) 2 =(x+y)(x+y) x+yx+y xxyy x2x2 xy y2y2 Multiply down the columns, then combine like terms

(x+y) 2 x 2 +2xy+y 2

(x+y) 2 (x+y) 2 =x 2 +2xy+y 2 What is the degree of the expansion? How many terms are in the expansion? What is the exponent of x in the first term? What is the exponent of y in the first term? What is the sum of the exponents in the first term?

(x+y) 2 (x+y) 2 =x 2 +2xy+y 2 What is the exponent of x in the second term? What is the exponent of y in the second term? What is the sum of the exponents in the second term? What is the exponent of x in the third term? What is the exponent of y in the third term? What is the sum of the exponents in the third term?

(x+y) 2 (x+y) 2 =x 2 +2xy+y 2 How do the exponents of x change from left to right? How do the exponents of y change from left to right? What is the coefficient of the first term? What is the coefficient of the second term? What is the coefficient of the third term?

(x+y) 3 (x+y) 3 =(x+y) 2 (x+y)

(x+y) 3 =(x 2 +2xy+y 2 )(x+y) x2x2 +2xy+y2y2 x2x2 + +y2y2 Write the first expression twice for the two terms in the second expression

(x+y) 3 =(x 2 +2xy+y 2 )(x+y) x2x2 +2xy+y2y2 x2x2 + +y2y2 xxxyyy Place each term of the second expression below

(x+y) 3 =(x 2 +2xy+y 2 )(x+y) x2x2 +2xy+y2y2 x2x2 + +y2y2 xxxyyy x3x3 2x 2 yxy 2 x2yx2y2xy 2 y3y3 Multiply down the columns, then combine like terms

(x+y) 3 (x+y) 3 =x 3 +3x 2 y+3xy 2 +y 3

(x+y) 3 What is the degree of the expansion? How many terms are in the expansion? What is the exponent of x in the first term? What is the exponent of y in the first term? What is the sum of the exponents in the first term? What is the exponent of x in the second term? What is the exponent of y in the second term? What is the sum of the exponents in the second term? (x+y) 3 =x 3 +3x 2 y+3xy 2 +y 3

(x+y) 3 What is the exponent of x in the third term? What is the exponent of y in the third term? What is the sum of the exponents in the third term? What is the exponent of x in the fourth term? What is the exponent of y in the fourth term? What is the sum of the exponents in the fourth term? (x+y) 3 =x 3 +3x 2 y+3xy 2 +y 3

(x+y) 3 How do the exponents of x change from left to right? How do the exponents of y change from left to right? What is the coefficient of the first term? What is the coefficient of the second term? What is the coefficient of the third term? (x+y) 3 =x 3 +3x 2 y+3xy 2 +y 3

(x+y) 4 (x+y) 4 =(x+y) 3 (x+y)

(x+y) 4 =(x 3 +3x 2 y+3xy 2 +y 3 )(x+y) x3x3 +3x 2 y+3xy 2 +y3y3 x3x3 +3x 2 y+3xy 2 +y3 Write the first expression twice for the two terms in the second expression

(x+y) 4 =(x 3 +3x 2 y+3xy 2 +y 3 )(x+y) x3x3 +3x 2 y+3xy 2 +y3y3 x3x3 +3x 2 y+3xy 2 +y3 xxxxyyyy Place each term of the second expression below

(x+y) 4 =(x 3 +3x 2 y+3xy 2 +y 3 )(x+y) x3x3 +3x 2 y+3xy 2 +y3y3 x3x3 +3x 2 y+3xy 2 +y3 xxxxyyyy x4x4 3x 3 y3x 2 y 2 xy 3 x3yx3y3x 2 y 2 3xy 3 y4y4 Multiply down the columns, then combine like terms

(x+y) 4 (x+y) 4 =x 4 +4x 3 y+6x 2 y 2 +4xy 3 +y 4

(x+y) 4 What is the degree of the expansion? How many terms are in the expansion? What is the exponent of x in the first term? What is the exponent of y in the first term? What is the sum of the exponents in the first term? What is the exponent of x in the second term? What is the exponent of y in the second term? What is the sum of the exponents in the second term? (x+y) 4 =x 4 +4x 3 y+6x 2 y 2 +4xy 3 +y 4

(x+y) 4 What is the exponent of x in the third term? What is the exponent of y in the third term? What is the sum of the exponents in the third term? What is the exponent of x in the fourth term? What is the exponent of y in the fourth term? What is the sum of the exponents in the fourth term? What is the exponent of x in the fifth term? What is the exponent of y in the fifth term? What is the sum of the exponents in the fifth term? (x+y) 4 =x 4 +4x 3 y+6x 2 y 2 +4xy 3 +y 4

(x+y) 4 How do the exponents of x change from left to right? How do the exponents of y change from left to right? What is the coefficient of the first term? What is the coefficient of the second term? What is the coefficient of the third term? What is the coefficient of the fourth term? What is the coefficient of the fifth term? (x+y) 4 =x 4 +4x 3 y+6x 2 y 2 +4xy 3 +y 4

Pattern of exponents degrees 1 to 4 ndegree# terms sum of exponents degree of expansion of binomial = n number of terms in expansion = n+1 sum of exponents in each term = n exponent of x decreases from n to 0 exponent of y increases from 0 to n

Pattern of coefficients degrees 1 to 4 degree coefficients What is the pattern from row to row?

Coefficients of 5 th degree expansion degree coefficients

This pattern of coefficients is called Pascal’s Triangle It can be extended to find the coefficients of any degree expansion of a binomial

(x+y) 5 What is the degree of the expansion? How many terms are in the expansion?

(x+y) 5 xy + xy + xy + xy + xy + xy What is the exponent of x in the first term? What is the exponent of y in the first term? What is the exponent of x in the second term? What is the exponent of y in the second term? What is the exponent of x in the third term? What is the exponent of y in the third term?

(x+y) 5 xy + xy + xy + xy + xy + xy What is the exponent of x in the fourth term? What is the exponent of y in the fourth term? What is the exponent of x in the fifth term? What is the exponent of y in the fifth term? What is the exponent of x in the sixth term? What is the exponent of y in the sixth term?

(x+y) 5 Based on the pattern for binomial coefficients: What is the binomial coefficient of the first term? What is the binomial coefficient of the second term? What is the binomial coefficient of the third term? What is the binomial coefficient of the fourth term? What is the binomial coefficient of the fifth term? What is the binomial coefficient of the sixth term?

(x+y) 5 = x 5 +5x 4 y+10x 3 y 2 +10x 2 y 3 +5xy 4 +y 5

Would you want to build Pascal’s Triangle for (x+y) 99 ? You could, but it would be a large triangle. Is there a short cut? Yes, indeed there is!

Factorials

n! Unary operator Symbol ! Multiplication of all numbers from n down to 1

0!=1 n!=n·(n-1)!=n·(n-1)·(n-2)! n! / (n-2)! = n·(n-1)·(n-2)! / (n-2)! =n·(n-1) ( n r ) means n choose r = n! / (n-r)!r!

(x+y) n Binomial Theorem For r = 0 to n The (r+1)th term is n! / (n-r)!r! x (n-r) y r

(x+y) 7 n=7 r=0 0+1=1 st term 7! / (7-0)!0! x (7-0) y 0 = 7! / 7! x 7 y 0 = x 7

(x+y) 7 n=7 r=1 1+1=2 nd term 7! / (7-1)!1! x (7-1) y 1 = 7! / 6! x 6 y 1 = 7·6! / 6! x 6 y 1 = 7x 6 y

(x+y) 7 n=7 r=2 2+1=3 rd term 7! / (7-2)!2! x (7-2) y 2 = 7! / 5!2! x 5 y 2 = 7·6·5! / 5!2! x 5 y 2 = 7·6 / 2 x 5 y 2 = 7·3x 5 y 2 = 21x 5 y 2

(x+y) 7 n=7 r=3 3+1=4 th term 7! / (7-3)!3! x (7-3) y 3 = 7! / 4!3! x 4 y 3 = 7·6·5·4! / 4!3! x 4 y 3 = 7·6·5 / 3·2·1 x 4 y 3 = 7·5x 4 y 3 = 35x 4 y 3

(x+y) 7 n=7 r=4 4+1=5 th term 7! / (7-4)!4! x (7-4) y 4 = 7! / 3!4! x 3 y 4 = 7·6·5·4! / 3!4! x 3 y 4 = 7·6·5 / 3·2·1 x 3 y 4 = 7·5x 3 y 4 = 35x 3 y 4

(x+y) 7 n=7 r=5 5+1=6 th term 7! / (7-5)!5! x (7-5) y 5 = 7! / 2!5! x 2 y 5 = 7·6·5! / 2!5! x 2 y 5 = 7·6 / 2·1 x 2 y 5 = 7·3x 2 y 5 = 21x 2 y 5

(x+y) 7 n=7 r=6 6+1=7 th term 7! / (7-6)!6! x (7-6) y 6 = 7! / 1!6! x 1 y 6 = 7·6! / 6! x 1 y 6 = 7xy 6

(x+y) 7 n=7 r=7 7+1=8 th term 7! / (7-7)!7! x (7-7) y 7 = 7! / 0!7! x 0 y 7 = 7! / 7! x 0 y 7 = y 7

(x+y) 7 x 7 +7x 6 y+21x 5 y 2 +35x 4 y x 3 y 4 +21x 2 y 5 +7xy 6 +y 7

What about x + a number

(x+6) 4 = x 4 +4x 3 (6) 1 +6x 2 (6) 2 +4x(6) 3 +(6) 4 = x 4 +24x x x+1296

Knowing the binomial theorem Can help you factor polynomials

8x 3 +36x 2 +54x+27 (x+y) 3 =x 3 +3x 2 y+3xy 2 +y 3 Is the first term a cube?  8x 3 = (2x) 3 Is the fourth term a cube?  27=(3) 3 Divide the second term by the binomial coefficient 3  Is the coefficient the cube root of the first term squared times the cube root of the fourth term?  36x 2  3=12x 2 =(4x 2 )(3)=(2x) 2 (3) Divide the third term by the binomial coefficient 3  Is the coefficient the cube root of the first term times the cube root of the fourth term squared?  54x  3=18x=(2x)(9)=(2x)(3) 2 The polynomial is the cube of a binomial 8x 3 +36x 2 +54x+27=(2x+3) 3

Conclusions The degree of the expansion is the exponent the binomial is raised to The number of terms in the expansion is one more than the degree of the expansion Pascal’s Triangle gives the binomial coefficients of the expansion The binomial theorem is useful for large expansions

Questions