This is “Binomial Theorem”, section 9.4 from the book Advanced Algebra (v. 1.0). For details on it (including licensing), click here.

For more information on the source of this book, or why it is available for free, please see the project's home page. You can browse or download additional books there. You may also download a PDF copy of this book (41 MB) or just this chapter (1 MB), suitable for printing or most e-readers, or a .zip file containing this book's HTML files (for use in a web browser offline).

Has this book helped you? Consider passing it on:
Creative Commons supports free culture from music to education. Their licenses helped make this book available to you.
DonorsChoose.org helps people like you help teachers fund their classroom projects, from art supplies to books to calculators.

9.4 Binomial Theorem

Learning Objectives

  1. Evaluate expressions involving factorials.
  2. Calculate binomial coefficients.
  3. Expand powers of binomials using the binomial theorem.

Factorials and the Binomial Coefficient

We begin by defining the factorialThe product of all natural numbers less than or equal to a given natural number, denoted n!. of a natural number n, denoted n!, as the product of all natural numbers less than or equal to n.

n!=n(n1)(n2)321

For example,

7!=7654321=5,040Sevenfactorial5!=54321=120Fivefactorial3!=321=6Threefactorial1!=1=1Onefactorial

We define zero factorialThe factorial of zero is defined to be equal to 1; 0!=1. to be equal to 1,

0!=1Zerofactorial

The factorial of a negative number is not defined.

 

Note: On most modern calculators you will find a factorial function. Some calculators do not provide a button dedicated to it. However, it usually can be found in the menu system if one is provided.

The factorial can also be expressed using the following recurrence relation,

n!=n(n1)!

For example, the factorial of 8 can be expressed as the product of 8 and 7!:

8!=87!=87654321=40,320

When working with ratios involving factorials, it is often the case that many of the factors cancel.

Example 1

Evaluate: 12!6!.

Solution:

12!6!=121110987654321654321=1211109876!6!=121110987=665,280

Answer: 665,280

The binomial coefficientAn integer that is calculated using the formula: (nk)=n!k!(nk)!., denoted nCk=(nk), is read “n choose k” and is given by the following formula:

nCk=(nk)=n!k!(nk)!

This formula is very important in a branch of mathematics called combinatorics. It gives the number of ways k elements can be chosen from a set of n elements where order does not matter. In this section, we are concerned with the ability to calculate this quantity.

Example 2

Calculate: (73).

Solution:

Use the formula for the binomial coefficent,

(nk)=n!k!(nk)!

where n=7 and k=3. After substituting, look for factors to cancel.

(73)=7!3!(73)!=7!3!4!=7654!3!4!=2106=35

Answer: 35

Note: Check the menu system of your calculator for a function that calculates this quantity. Look for the notation nCk in the probability subsection.

Try this! Calculate: (85).

Answer: 56

Consider the following binomial raised to the 3rd power in its expanded form:

(x+y)3=x3+3x2y+3xy2+y3

Compare it to the following calculations,

(30)=3!0!(30)!=3!13!=1(31)=3!1!(31)!=32!12!=3(32)=3!2!(32)!=32!2!1!=3(33)=3!3!(33)!=3!3!0!=1

Notice that there appears to be a connection between these calculations and the coefficients of the expanded binomial. This observation is generalized in the next section.

Binomial Theorem

Consider expanding (x+2)5:

(x+2)5=(x+2)(x+2)(x+2)(x+2)(x+2)

One quickly realizes that this is a very tedious calculation involving multiple applications of the distributive property. The binomial theoremDescribes the algebraic expansion of binomials raised to powers: (x+y)n=Σk=0n(nk)xnkyk. provides a method of expanding binomials raised to powers without directly multiplying each factor:

(x+y)n=(n0)xny0+(n1)xn1y1+(n2)xn2y2++(nn1)x1yn1+(nn)x0yn

More compactly we can write,

(x+y)n=Σk=0n(nk)xnkykBinomialtheorem

Example 3

Expand using the binomial theorem: (x+2)5.

Solution:

Use the binomial theorem where n=5 and y=2.

(x+2)5=(50)x520+(51)x421+(52)x322+(53)x223+(54)x124+(55)x025

Sometimes it is helpful to identify the pattern that results from applying the binomial theorem. Notice that powers of the variable x start at 5 and decrease to zero. The powers of the constant term start at 0 and increase to 5. The binomial coefficents can be calculated off to the side and are left to the reader as an exercise.

(x+2)5=(50)x520+(51)x421+(52)x322+(53)x223+(54)x124+(55)x025=1x5×1+5x4×2+10x3×4+10x2×8+5x1×16+1×1×32=x5+10x4+40x3+80x2+80x+32

Answer: x5+10x4+40x3+80x2+80x+32

The binomial may have negative terms, in which case we will obtain an alternating series.

Example 4

Expand using the binomial theorem: (u2v)4.

Solution:

Use the binomial theorem where n=4, x=u, and y=2v and then simplify each term.

(u2v)4=(40)u4(2v)0+(41)u3(2v)1+(42)u2(2v)2+(43)u1(2v)3+(44)u0(2v)4=1×u4×1+4u3(2v)+6u2(4v2)+4u(8v3)+1×1(16v4)=u48u3v+24u2v232uv3+16v4

Answer: u48u3v+24u2v232uv3+16v4

Try this! Expand using the binomial theorem: (a23)4.

Answer: a812a6+54a4108a2+81

Next we study the coefficients of the expansions of (x+y)n starting with n=0:

(x+y)0=1(x+y)1=x+y(x+y)2=x2+2xy+y2(x+y)3=x3+3x2y+3xy2+y3(x+y)4=x4+4x3y+6x2y2+4xy3+y4

Write the coefficients in a triangular array and note that each number below is the sum of the two numbers above it, always leaving a 1 on either end.

This is Pascal’s triangleA triangular array of numbers that correspond to the binomial coefficients.; it provides a quick method for calculating the binomial coefficients. Use this in conjunction with the binomial theorem to streamline the process of expanding binomials raised to powers. For example, to expand (x1)6 we would need two more rows of Pascal’s triangle,

The binomial coefficients that we need are in blue. Use these numbers and the binomial theorem to quickly expand (x1)6 as follows:

(x1)6=1x6(1)0+6x5(1)1+15x4(1)2+20x3(1)3+15x2(1)4+6x(1)5+1x0(1)6=x66x5+15x420x3+15x26x+1

Example 5

Expand using the binomial theorem and Pascal’s triangle: (2x5)4.

Solution:

From Pascal’s triangle we can see that when n=4 the binomial coefficients are 1, 4, 6, 4, and 1. Use these numbers and the binomial theorem as follows:

(2x5)4=1(2x)4(5)0+4(2x)3(5)1+6(2x)2(5)2+4(2x)1(5)3+1(2x)0(5)4=16x41+48x3(5)+64x225+42x(125)+1625=16x4160x3+600x21,000x+625

Answer: 16x4160x3+600x21,000x+625

Key Takeaways

  • To calculate the factorial of a natural number, multiply that number by all natural numbers less than it: 5!=54321=120. Remember that we have defined 0!=1.
  • The binomial coefficients are the integers calculated using the formula: (nk)=n!k!(nk)!.
  • The binomial theorem provides a method for expanding binomials raised to powers without directly multiplying each factor: (x+y)n=Σk=0n(nk)xnkyk.
  • Use Pascal’s triangle to quickly determine the binomial coefficients.

Topic Exercises

    Part A: Factorials and the Binomial Coefficient

      Evaluate.

    1. 6!

    2. 4!

    3. 10!

    4. 9!

    5. 6!3!

    6. 8!4!

    7. 13!9!

    8. 15!10!

    9. 12!3!7!

    10. 10!2!5!

    11. n!(n2)!

    12. (n+1)!(n1)!

      1. 4!+3!
      2. (4+3)!
      1. 4!3!
      2. (43)!

      Rewrite using factorial notation.

    1. 1×2×3×4×5×6×7

    2. 1×2×3×4×5

    3. 15×14×13

    4. 10×9×8×7

    5. 13

    6. 8×7

    7. n(n1)(n2)

    8. 1×2×3××n×(n+1)

      Calculate the indicated binomial coefficient.

    1. (64)
    2. (84)
    3. (72)
    4. (95)
    5. (90)
    6. (1312)
    7. (n0)
    8. (nn)
    9. (n1)
    10. (nn1)
    11. 10C8

    12. 5C1

    13. 12C12

    14. 10C5

    15. nCn2

    16. nCn3

    Part B: Binomial Theorem

      Expand using the binomial theorem.

    1. (4x3)3

    2. (2x5)3

    3. (x2+y)3

    4. (x+1y)3

    5. (x+3)4

    6. (x+5)4

    7. (x4)4

    8. (x2)4

    9. (x+2y)4

    10. (x3y)4

    11. (x+1)5

    12. (x3)5

    13. (x2)6

    14. (x+1)6

    15. (x1)7

    16. (x+1)7

    17. (5x1)4

    18. (3x2)4

    19. (4u+v)4

    20. (3uv)4

    21. (u5v)5

    22. (2u+3v)5

    23. (ab2)5

    24. (a2+b2)4

    25. (a2+b4)6

    26. (a5+b2)5

    27. (x+2)3

    28. (x2)4

    29. (xy)4, x,y0

    30. (x+2y)5, x,y0

    31. (x+y)7

    32. (x+y)8

    33. (x+y)9

    34. (xy)7

    35. (xy)8

    36. (xy)9

    Part C: Discussion Board

    1. Determine the factorials of the integers 5, 10, 15, 20, and 25. Which grows faster, the common exponential function an=10n or the factorial function an=n!? Explain.

    2. Research and discuss the history of the binomial theorem.

Answers

  1. 720

  2. 3,628,800

  3. 120

  4. 17,160

  5. 15,840

  6. n2n

    1. 30
    2. 5,040
  7. 7!

  8. 15!12!

  9. 13!12!

  10. n!(n3)!

  11. 15

  12. 21

  13. 1

  14. 1

  15. n

  16. 45

  17. 1

  18. n2n2

  1. 64x3144x2+108x27

  2. x38+3x2y4+3xy22+y3

  3. x4+12x3+54x2+108x+81

  4. x416x3+96x2256x+256

  5. x4+8x3y+24x2y2+32xy3+16y4

  6. x5+5x4+10x3+10x2+5x+1

  7. x612x5+60x4160x3+240x2192x+64

  8. x77x6+21x535x4+35x321x2+7x1

  9. 625x4500x3+150x220x+1

  10. 256u4+256u3v+96u2v2+16uv3+v4

  11. u525u4v+250u3v21,250u2v3+3,125uv43,125v5
  12. a55a4b2+10a3b410a2b6+5ab8b10

  13. a12+6a10b4+15a8b8+20a6b12+15a4b16+6a2b20+b24
  14. x3+32x2+6x+22

  15. x24xxy+6xy4yxy+y2

  16. x7+7x6y+21x5y2+35x4y3+35x3y4+21x2y5+7xy6+y7
  17. x9+9x8y+36x7y2+84x6y3+126x5y4+126x4y5+84x3y6+36x2y7+9xy8+y9
  18. x88x7y+28x6y256x5y3+70x4y456x3y5+28x2y68xy7+y8
  1. Answer may vary