§ 11.4 The Binomial Theorem. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 11.4 The Binomial Coefficient In this section, we look at methods for.

§ 11.4

The Binomial Theorem

Blitzer, Intermediate Algebra, 5e – Slide #2 Section 11.4

The Binomial Coefficient

In this section, we look at methods for raising binomials to powers. For example, if we wished to cube the expression (x + 2), we could do long multiplication repeatedly. But what if we wished to raise (x + 2) to the 10th power? We need an efficient method for doing that. The Binomial Theorem provides just such a method for us.

We use the Binomial Theorem for computing powers of binomials. Another very useful tool that can be used which will be presented also in this section is Pascal’s Triangle.

Blitzer, Intermediate Algebra, 5e – Slide #3 Section 11.4

The Binomial Theorem

Properties of T1) The first term in the expansion of is . The exponents on a decrease by 1 in each successive term.

2) The exponents on b in the expansion of increase by 1 in each successive term. In the first term, the exponent on b is 0. The last term is .

3) The sum of the exponents on the variables in any term in the expansion of is equal to n.

4) The number of terms in the polynomial expansion is one greater than the power of the binomial, n. There are n + 1 terms in the expanded form of .

nba nba na

nba

nb

nba

nba

Blitzer, Intermediate Algebra, 5e – Slide #4 Section 11.4

The Binomial Coefficient

Definition of a Binomial Coefficient T

For nonnegative integers n and r, with , the expression

(read “n above r”) is called a binomial coefficient and is

defined by

r

n

rn

r

n

.! !

!

rnr

n

r

n

Blitzer, Intermediate Algebra, 5e – Slide #5 Section 11.4

The Binomial Coefficient

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Evaluate:

Apply the definition of the binomial coefficient.

. 2

15

105!1312

!131415

!13 !2

!15

! 215!2

!15

2

15

Blitzer, Intermediate Algebra, 5e – Slide #6 Section 11.4

The Binomial Theorem

A Formula for Expanding Binomials:

The Binomial TheoremFor any positive integer n,

nnnnnn bn

nba

nba

nba

na

nba

33221

3210

.0

n

r

rrn bar

n

Blitzer, Intermediate Algebra, 5e – Slide #7 Section 11.4

The Binomial Theorem

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Expand:

Because the Binomial Theorem involves the addition of two terms raised to a power, we rewrite as . We use the Binomial Theorem to expand . In this expression, , b = -1, and n = 4. In the expansion, powers of are in descending order, starting with . Powers of -1 are in ascending order, starting with . [Because , a -1 is not shown in the first term.] The sum of the exponents on and -1 in each term is equal to 4, the external exponent in the original expression.

.1245 x

45 12 x 45 12 x 45 12 x

52xa 52x 452x

01 11 0

52x

Blitzer, Intermediate Algebra, 5e – Slide #8 Section 11.4

The Binomial Theorem

4545 1212 xx

CONTINUECONTINUEDD

43345

224514545

14

412

3

4

122

412

1

42

0

4

x

xxx

43345

224514545

1!0 !4

!412

!1 !3

!4

12!2 !2

!412

!3 !1

!42

!4 !0

!4

x

xxx

Rewrite using the Binomial Theorem.

Use the definition of the binomial coefficient.

Blitzer, Intermediate Algebra, 5e – Slide #9 Section 11.4

The Binomial Theorem

CONTINUECONTINUEDD

43345224514545 1112412612421

xxxx

11124146184161 5101520 xxxx

18243216 5101520 xxxx

Evaluate all factorial expressions.

Evaluate all exponents.

Multiply.

Blitzer, Intermediate Algebra, 5e – Slide #10 Section 11.4

The Binomial Theorem

Finding a Particular Term in a Binomial Expansion

The (r + 1)st term of the expansion of is nba

.rrn bar

n

Blitzer, Intermediate Algebra, 5e – Slide #11 Section 11.4

The Binomial Theorem

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Find the sixth term in the expansion of

Since we are looking for the sixth term in the expansion, this is the (r + 1)st term Therefore, r + 1 = 6. So r = 5.

.823 yx

Since the exponent outside the parentheses on is 8, n = 8.

823 yx

Since the first term in is 823 yx . , 33 xax

Since the second term in is 823 yx . , 22 yby

Now we can use the formula to find the sixth term.rrn bar

n

Blitzer, Intermediate Algebra, 5e – Slide #12 Section 11.4

The Binomial Theorem

The sixth term is,

52583

5

8yxba

r

n rrn

CONTINUECONTINUEDD

5233

! 58!5

!8yx

10956 yx

In the formula, replace n with 8, r with 5, a with , and b with .

3x2y

Simplify.

Simplify.

Blitzer, Intermediate Algebra, 5e – Slide #13 Section 11.4

The Binomial Theorem – Pascal’s

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

Blitzer, Intermediate Algebra, 5e – Slide #14 Section 11.4

The Binomial Theorem – Pascal’s

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

Do you see the pattern?Start with a 1 at the top. Always put 1s on the diagonals. Get each number on a successive rows by adding the two numbers just above that number.Start at the top and work down. See if you can reproduce Pascal’s Triangle.

Blitzer, Intermediate Algebra, 5e – Slide #15 Section 11.4

Using Pascal’s Triangle

EXAMPLEEXAMPLE

SOLUTIONSOLUTION

Expand:

We will do this same problem that we worked earlier, but use Pascal’s triangle this time. First we note that we are raising a binomial to the 4th power. That tells us that we will have 5 terms in our answer. Each term in the answer will have powers of our two expressions in the binomial we started with. In the first term, we will have our first expression raised to the 4th and the second one to the 0. In the second term of our answer, we will have the first expression raised to the third with the second expression raised to the one. In the third term of our answer, we will have the first expression raised to the 2nd power and the second raised also to the 2nd. In the fourth expression, we will

.1245 x

Blitzer, Intermediate Algebra, 5e – Slide #16 Section 11.4

Using Pascal’s Triangle

The first expression raised to the 1st power and the second raised to the 3rd power. In the fifth term of our answer, we will have the first expression raised to the 0 and the second raised to the 4th. For each term of our answer, the sum of the powers we used is 4, the power that we originally raised the binomial to. Now – we must put coefficients on our terms. That’s where we use Pascal’s triangle. We need 5 coefficients. We look down to the 5th row. The coefficients we see there are: 1,4,6,4,1. Those are our magic coefficients. Note that we have 1s on the ends and that the second number and next to last number is the power we raised the binomial to. See how easy that was! Let’s put it all together now. Our answer is:

Blitzer, Intermediate Algebra, 5e – Slide #17 Section 11.4

Using Pascal’s Triangle

4545 1212 xx

CONTINUECONTINUEDD

4315

2253545

11124

12612421

x

xxx

Expand, writing the 5 terms you will get and putting in coefficients that you found from Pascal’s Triangle. Note that powers of the first expression in the binomial decrease as the other expression’s powers increase. Sum is always 4 here.

Now, just simplify and you have your answer. Easy enough? Note that the signs are alternating.

18243216 5101520 xxxx

Blitzer, Intermediate Algebra, 5e – Slide #18 Section 11.4

In Conclusion…

Raising binomials to powers can be a little messy – but we have formulas! We have the Binomial Theorem and we have Pascal’s Triangle. Most students find that using Pascal’s Triangle is quicker and easier to use than the Binomial Theorem. The exception would be when you are looking for just a single term in the expansion of a binomial to a very large power. For Pascal’s Triangle, you will just need to learn how to construct the triangle, and you will need to practice using it! And, as in all mathematics, the longer you practice – the easier it gets.