12.8. Binomial Theorem http://www.ck12.org
Be extremely careful when working with binomials of the form(a−b)n. You need to remember to capture the
negative with the second term as you write out the expansion:(a−b)n= (a+(−b))n.
Example A
Expand the following binomial using the Binomial Theorem.
(m−n)^6
Solution:
(m−n)^6 =(^60 )m^6 +(^61 )m^5 (−n)^1 +(^62 )m^4 (−n)^2 +(^63 )m^3 (−n)^3
+(^64 )m^2 (−n)^4 +(^65 )m^1 (−n)^5 +(^66 )(−n)^6
= 1 m^6 − 6 m^5 n+ 15 m^4 n^2 − 20 m^3 n^3 + 15 m^2 n^4 − 6 m^1 n^5 + 1 n^6Example B
What is the coefficient of the termx^7 y^9 in the expansion of the binomial(x+y)^16?
Solution:The Binomial Theorem allows you to calculate just the coefficient you need.
( 16
9
)= 16!
9!7!=^16 ·^157 ··^146 · 5 ··^134 ·· 312 · 2 ··^111 ·^10 =^11 ,^440
Example C
What is the coefficient ofx^6 in the expansion of( 4 − 3 x)^7?
Solution:For this problem you should calculate the whole term, since the 3 and the 4 in( 3 − 4 x)will impact the
coefficient ofx^6 as well.(^76 ) 41 (− 3 x)^6 = 7 · 4 · 729 x^6 = 20 , 412 x^6. The coefficient is 20,412.
Concept Problem Revisited
The expanded version of( 2 x− 3 )^5 is:
( 2 x− 3 )^5 =(^50 )( 2 x)^5 +(^51 )( 2 x)^4 (− 3 )^1 +(^52 )( 2 x)^3 (− 3 )^2
+(^53 )( 2 x)^2 (− 3 )^3 +(^54 )( 2 x)^1 (− 3 )^4 +(^55 )(− 3 )^6
= ( 2 x)^5 + 5 ( 2 x)^4 (− 3 )^1 + 10 ( 2 x)^3 (− 3 )^2
+ 10 ( 2 x)^2 (− 3 )^3 + 5 ( 2 x)^1 (− 3 )^4 +(− 3 )^5
= 32 x^5 − 240 x^4 + 720 x^3 − 1080 x^2 + 810 x− 243