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6.5 Binomial Theorem

Previously, we learned patterns for squaring and cubing binomials.

(x+y)2= x2 + 2xy + y2 (x+y)3= x3 + 3x2y + 3xy2 + y3

If the patterns were not memorized, you could simply write it out and multiply.

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

The square of a binomial is not a problem, just foil. The cube of a binomial is a little work. But any exponent higher than 3 is a problem. That would just be too much work to multiply 4 binomials.

(x+y)4= (x + y)(x + y)(x + y)(x + y) Too much work! Agreed?

Fortunately we have an alternative method which is the Binomial Theorem. Before we learn the binomial theorem, we need to have a little fun playing with our scientific calculators. We will only show what you need to “get by” on this section. We will not get involved with factorials, permutations, and combinations. We would give more detail on these topics in a course covering probability and statistics.

n

r

The notation: is read n choose r, “combinations of n things taken r at a time.”

For example if we had 6 people, how many different groups of 2 could we choose?

We can calculate this on a scientific calculator. We will show how to perform this operation on a TI-30XIIS and a graphing calculator. If you calculator different, please go to an assistant or your instructor.

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6.5 Binomial Theorem

1. Type in the 10.2. Press the PRB key.3. The choices are nPr, nCr, and !. Use the arrows

to underline the nCr then press enter. 4. Type in the 7.5. Press enter and you should get 120.

1st the TI-30XIIS. To find 10

7

2nd, a graphing calculator.

To find 10

7 1. Type in the 10.

2. Press the math key. Use the arrow to highlight PRB. Then scroll down to the nCr and press enter.

3. Then type in the 7 and press enter. You should still get 120.

Examples.

12792

5

242024

3

111

11

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6.5 Binomial Theorem

2) For each term, the sum of the exponents on the x and y expressions is n.

1) The powers of x are decreasing while the powers of y are increasing.

3) The top number in the binomial coefficient always equals n.

4) For each term, the exponent on the y expression and bottom number in the binomial coefficient are always one less than the number of the term.

Things to note:

n n 0 n 1 1 n 2 2 0 nn n n nx + y = x y x y x y x y

0 1 2 n

The Binomial Theorem

For algebraic expressions, x and y, and any natural number, n,

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6.5 Binomial Theorem

Example 1: Use the binomial theorem to multiply (expand): (x + y)6

6 0 5 1 4 2 3 3 2 4 1 5 0 66 6 6 6 6 6 6

x y x y x y x y x y x y x y0 1 2 3 4 5 6

6 5 4 2 3 3 2 4 5 6x 6x y 15x y 20x y 15x y 6xy y

Your Turn Problem #1

Expand (x + y)8

7 6 2 5 3 3 5 2 6 7 88 4 4 x 8x y 28x y 56x y 70x y 56x y 28x y 8xy y

Answer:

6 0 5 1 4 2 3 3 2 4 1 5 0 61 x y 6 x y 15 x y 20 x y 15 x y 6 x y 1 x y

n = 6

n 0 0 n 1 1 n 2 2 0 nn n n nx y x y x y x y

0 1 2 n

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6.5 Binomial Theorem

5 0 4 1 3 2 2 3 1 4 0 55 5 5 5 5 5x 7 x 7 x 7 x 7 x 7 x 7

0 1 2 3 4 5

5 4 3 2 1 01 x 5 x (7) 10 x (49) 10 x (343) 5 x (2401) 1 x 16807

Example 2. Use the binomial theorem to multiply (expand): (x + 7)5

Note: the x and y may not be variables. They can be any algebraic expression.

5 4 3 2x 35x 490x 3430x 12005x 16807

(x + 7)5 ; (substitute 7 for y in the binomial expansion)

Your Turn Problem #2

Expand (x + 5)7

Answer:

7 6 5 4 3 2x 35x 525x 4375x 21875x 65625x 109375x 78125

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6.5 Binomial Theorem

2 5 3 0 2 4 3 1 2 3 3 2 2 2 3 3 2 1 3 4 2 0 3 55 5 5 5 5 5

(5a ) (4b ) (5a ) (4b ) (5a ) (4b ) (5a ) (4b ) (5a ) (4b ) (5a ) (4b ) 0 1 2 3 4 5

10 8 3 6 6 4 9 2 12 153125a 5(625a )(4b ) 10(125a )(16b ) 10(25a )(64b ) 5(5a )(256b ) 1024b

Example 3. Use the binomial theorem to multiply (expand): (5a2 + 4b3)5

Note: the x and y can represent terms with variable powers.

10 8 3 6 6 4 9 2 12 153125a 12500a b 20000a b 16000a b 6400a b 1024b

(5a2+ 4b3)5 ; (substitute 5a2 for x and 4b3 for y in the binomial expansion)

Your Turn Problem #3

Expand (9m5 + 3k2)4

Answer:

20 15 2 10 4 5 6 86561m 8748m k 4374m k 972m k 81k

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6.5 Binomial Theorem

3 7 0 3 6 1 3 5 2 3 4 3 3 3 4 3 2 57 7 7 7 7 7

(2p ) ( 5) (2p ) ( 5) (2p ) ( 5) (2p ) ( 5) (2p ) ( 5) (2p ) ( 5)0 1 2 3 4 5

3 1 6 3 0 77 7 (2p ) ( 5) (2p ) ( 5)

6 7

21 18 15 12 9 6

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128p 7(64p )( 5) 21(32p )(25) 35(16p )( 125) 35(8p )(625) 21(4p )( 3125)

7(2p )(15625) ( 78125)

Example 4. Use the binomial theorem to multiply (expand): (2p3 – 5)7

Note: the terms can have negative signs.

(2p3 – 5)7 ; (substitute 2p3 for x and -5 for y in the binomial expansion)

21 18 15 12 9 6 3 128p 2240p 16800p 70000p 175000p 262500p 218750p 78125

(note: since one line of these problems is a little long, you may want to turn your paper sideways.)

Your Turn Problem #4

Expand (4n2 – 7)6

Answer:

12 10 8 6 4 2 4096n 43008n 188160n 439040n 576240n 403368n 117649

The End

4-26-07