1 Special products (Section P.3 , page 34)
2 Factorials (Section 8.1 , page 721)
3 Summation notation (Section 8.1 , page 722)
Blaise Pascal was a French mathematician. An evident genius, he was 14 when he began to accompany his father to gatherings of mathematicians that were arranged by Father Marin Mersenne. At the age of 16, Pascal presented his own results at one of Mersenne’s meetings.
Pascal’s desire to help his father with his work collecting taxes led him to invent the first digital calculator, called the Pascaline. Pascal was also interested in atmospheric pressure, and in 1648, he observed that the pressure of the atmosphere decreased with height and that a vacuum existed above the atmosphere.
Pascal’s intense interest in mathematics led him to produce important results related to conic sections and to engage in correspondence with Fermat in which he formulated the foundations for the theory of probability. Pascal died a painful death from cancer at the age of 39.
In this section, we study Pascal’s Triangle. This triangle of numbers contains the important binomial coefficients. See page 760. Pascal’s work was influential in Newton’s discovery of the general Binomial Theorem for fractional and negative powers.
Recall that a polynomial that has exactly two terms is called a binomial. In the current section, we study a method for expanding (x+y)n
First, consider these binomial expansions:
The expansions of (x+y)2
The patterns for expansions of (x+y)n
The expansion of (x+y)n
The sum of the exponents on x and y in each term equals n.
The exponent on x starts at n(xn=xn⋅y0)
The exponent on y starts at 0 (xn=xn⋅y0)
The variables x and y have symmetric roles. That is, replacing x with y and y with x in the expansion of (x+y)n
You may also have noticed that the coefficients of the first and last terms are both 1 and the coefficients of the second and the next-to-last terms are equal. In general, the coefficients of
are equal for j=0, 1, 2, …, n.
The coefficients in the expansion of (x+y)n
1 Use Pascal’s Triangle to compute binomial coefficients.
As early as a.d. 1100, the Chinese scholar Chia Hsien had discovered the secret of the binomial coefficients that was later rediscovered by the French philosopher and mathematician Blaise Pascal (1623–1662). To understand Chia Hsien’s and Pascal’s construction of binomial coefficients, let’s first look at the coefficients in these binomial expansions:
If we remove the variables and the plus signs and list only the coefficients, we get a triangle of numbers composed of the binomial coefficients. This triangle of numbers is known as Pascal’s Triangle.
Note the symmetry in Pascal’s Triangle. If the triangle were folded vertically down the middle, the numbers on each side of the crease would match. To create a new bottom row in the triangle, put the number 1 in the first and last places of the new row and add two neighboring entries in the previous row.
The top row is called the zeroth row because it corresponds to the binomial expansion of (x+y)0.
2 Use Pascal’s Triangle to expand a binomial power.
Expand (4y−2x)5.
From the fifth row of Pascal’s Triangle, we see that the binomial coefficients are
We must make some changes in the expansion
to get the expansion for (4y−2x)5:
Replace x with 4y.
Replace y with −2x.
We note that expanding a difference results in alternating signs between terms.
Expand (3y−x)6.
3 Use the Binomial Theorem to expand a binomial power.
The coefficients in a binomial expansion can be computed by using ratios of certain factorials.
We first introduce the symbol (nr)
The symbol (nr)
Evaluate each expression.
(41)
(53)
(90)
(3535)
(41)=4!1!(4−1)!=4!1! 3!=4⋅3⋅2⋅11(3⋅2⋅1)=41=4
(53)=5!3!(5−3!)=5!3! 2!=5⋅4⋅3!3! 2!=5⋅42=5⋅2=10
(90)=9!0!(9−0)!=9!0! 9!=11=1Recall that 0!=1.
(3535)=35!35!(35−35)!=35!35! 0!=11=1
Evaluate each expression.
(62)
(129)
The numbers (nr)
or
This result is the Binomial Theorem (for n=4
The kth term in the expansion of (x+y)n
Find the binomial expansion of (x−3y)4.
Replace y with −3y
Find the binomial expansion of (3x−y)4.
4 Find the coefficient of a term in a binomial expansion.
The Binomial coefficients (nr)
Find the coefficient of x9y3
The coefficient of xn−ryr is (nr)
Find the coefficient of x3y9
The method illustrated in Example 4 allows us to find any particular term in a binomial expansion without writing out the complete expansion.
Find the term containing x10
We begin with the formula for the term containing the factor xr.
Find the term containing x3
Recall that, assuming decreasing powers of x, the kth term in the expansion of (x+y)n
Find the 15th term in the expansion of (2x−1)18
The kth term in the expansion of (x+y)n
The 15th term in the expansion of (2x−1)18
Find the fourth term in the expansion of (x−3)12
The expansion of (x+y)n
In the expansion of (x+y)5,
Expanding a difference such as (2x−y)10
For any positive integer n, (nn)=_
True or False. Every coefficient in the expansion of (x+y)n,
True or False. If n>3,
True or False. If n≥1
True or False. The coefficient of the sixth term in the expansion of (x+y)55
In Exercises 9–24, evaluate each expression.
6!3!
11!9!
12!11!
3!0!
(64)
(63)
(90)
(120)
(71)
(73)
(4545)
(450)
(10098)
(1002)
(2119)
(152)
In Exercises 25–32, use Pascal’s Triangle to expand each binomial.
(x+2)4
(x+3)4
(x−2)5
(3−x)5
(2−3x)3
(3−2x)3
(2x+3y)4
(2x+5y)4
In Exercises 33–54, use either the Binomial Theorem or Pascal’s Triangle to expand each binomial.
(x+1)4
(x+2)4
(x−1)5
(1−x)5
(y−3)3
(2−y)5
(x+y)6
(x−y)6
(1+3y)5
(2x+1)5
(2x+1)4
(3x−1)4
(x−2y)3
(2x−y)3
(2x+y)4
(3x−2y)4
(x2+2)7
(2−x2)7
(a2−13)4
(12−a2)4
(1x+y)3
(x+2y)3
In Exercises 55–62, find the specified coefficient.
for x2y7
for x10y5
for x5
for x7
for x2y7
for x3y4
for x10y12
for x18y6
In Exercises 63–76, find the specified term. In Exercises 71–76 assume descending powers of x.
(x+y)10;
(x+y)10;
(x−2)12;
(2−x)12;
(2x+3y)8;
(2x+3y)8;
(5x−2y)11;
(7x−y)15;
(x−1)9
(x−1)9
(2x−y)7
(−2x+y)7
(x+3y)10
(x+0.5y)8
Find the value of (1.2)5
Use the method in Exercise 77 to evaluate each expression.
(2.9)4
(10.4)3
Find the middle term in the expansion of (√x−2x2)10.
Find the middle term in the expansion of (√x+1x2)10.
Find the middle term in the expansion of (1−x2y−3)12.
Show that the middle term in the expansion of (1+x)2n
Prove that (n0)+(n1)+(n2)+⋯+(nn)=2n.
Prove that (n0)−(n1)+(n2)−(n3)+⋯+(−1)n(nn)=0.
Prove that (kj)+(kj−1)=(k+1j).
Prove the Binomial Theorem by using the principle of mathematical induction.
[Hint:
Use Exercise 85 to show that
In Exercises 87–89, find the value of the expression without expanding any term.
(2x−1)4+4(2x−1)3(3−2x)+6(2x−1)2(3−2x)2+4(2x−1)(3−2x)3+(3−2x)4
(x+1)4−4(x+1)3(x−1)+6(x+1)2(x−1)2−4(x+1)(x−1)3+(x−1)4
(3x−1)5+5(3x−1)4(1−2x)+10(3x−1)3(1−2x)2+10(3x−1)2(1−2x)3+5(3x−1)(1−2x)4+(1−2x)5
Find the constant term in each expansion.
(x2−1x)9
(√x−2x2)10
If the constant term in the expansion of (kx−1x2)6
If the constant term in the expansion of (x3+kx8)11
Show that there is no constant term in the expansion of (2x2−14x)11.
Use the binomial expansion of (x+y)6, with x=1 and y=1, to show that
Use the binomial expansion of (x+y)10 to show that
Use the binomial expansion of (x+y)2 to show that if x>0 and y>0, then (x+y)2>x2+y2.
Use the binomial expansion of (x+y)n to show that if x>0 and y>0, then (x+y)n>xn+yn.
Use the binomial expansion of (1+x)n to show that if x>0, then (1+x)n>1+nx.
In Exercises 99–106, write each expression using factorial notation.
5⋅6⋅7⋅8
10⋅9⋅8⋅7⋅6
2⋅4⋅6⋅8⋅10⋅12
3⋅6⋅9⋅12⋅15
Compute 12!10!.
Compute n!(n−r)! for n=100 and r=2.
Compute 8!5!3!.
Compute n!(n−r)!r! for n=10 and r=3.