http://www.ck12.org Chapter 12. Discrete Math
Vocabulary
TheBinomial Theoremis a theorem that states how to expand binomials that are raised to a power using combina-
tions. TheBinomial Theoremis:
(a+b)n=
n
i=∑ 0 (ni)aibn−iGuided Practice
- What is the coefficient ofx^3 in the expansion of(x− 4 )^5?
- Compute the following summation.
4
i∑= 0
( 4
i)
- Collapse the following polynomial using the Binomial Theorem.
32 x^5 − 80 x^4 + 80 x^3 − 40 x^2 + 10 x− 1
Answers:
1.(^52 )· 13 (− 4 )^2 = 160 - This is asking for(^40 )+(^41 )+···+(^44 ), which are the sum of all the coefficients of(a+b)^4.
1 + 4 + 6 + 4 + 1 = 16 - Since the last term is -1 and the power on the first term is a 5 you can conclude that the second half of the binomial
is(?− 1 )^5. The first term is positive and( 2 x)^5 = 32 x^5 , so the first term in the binomial must be 2x. The binomial is
( 2 x− 1 )^5.
Practice
Expand each of the following binomials using the Binomial Theorem.
1.(x−y)^4
2.(x− 3 y)^5
3.( 2 x+ 4 y)^7
- What is the coefficient ofx^4 in(x− 2 )^7?
- What is the coefficient ofx^3 y^5 in(x+y)^8?
- What is the coefficient ofx^5 in( 2 x− 5 )^6?
- What is the coefficient ofy^2 in( 4 y− 5 )^4?
- What is the coefficient ofx^2 y^6 in( 2 x+y)^8?
- What is the coefficient ofx^3 y^4 in( 5 x+ 2 y)^7?
Compute the following summations.
10.i∑=^90 (^9 i)
11.i^12 ∑= 0 (^12 i)
8
i∑= 0( 8
i