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§ 11.4 The Binomial Theorem

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**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 #2 Section 11.4

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**The Binomial Theorem Properties of T**

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

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**Definition of a Binomial Coefficient T**

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

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**The Binomial Coefficient**

EXAMPLE Evaluate: SOLUTION Apply the definition of the binomial coefficient. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 11.4

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**A Formula for Expanding Binomials:**

The Binomial Theorem A Formula for Expanding Binomials: The Binomial Theorem For any positive integer n, Blitzer, Intermediate Algebra, 5e – Slide #6 Section 11.4

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**The Binomial Theorem Expand:**

EXAMPLE Expand: SOLUTION 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. Blitzer, Intermediate Algebra, 5e – Slide #7 Section 11.4

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**The Binomial Theorem Rewrite using the Binomial Theorem.**

CONTINUED Rewrite using the Binomial Theorem. Use the definition of the binomial coefficient. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 11.4

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**The Binomial Theorem Evaluate all factorial expressions.**

CONTINUED Evaluate all factorial expressions. Evaluate all exponents. Multiply. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 11.4

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**Finding a Particular Term in a Binomial Expansion**

The Binomial Theorem Finding a Particular Term in a Binomial Expansion The (r + 1)st term of the expansion of is Blitzer, Intermediate Algebra, 5e – Slide #10 Section 11.4

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**The Binomial Theorem Find the sixth term in the expansion of**

EXAMPLE Find the sixth term in the expansion of SOLUTION 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. Since the exponent outside the parentheses on is 8, n = 8. Since the first term in is Since the second term in is Now we can use the formula to find the sixth term. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 11.4

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**The Binomial Theorem The sixth term is,**

CONTINUED The sixth term is, In the formula, replace n with 8, r with 5, a with , and b with Simplify. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #12 Section 11.4

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**The Binomial Theorem – Pascal’s**

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

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**The Binomial Theorem – Pascal’s**

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

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**Using Pascal’s Triangle**

EXAMPLE Expand: SOLUTION 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 Blitzer, Intermediate Algebra, 5e – Slide #15 Section 11.4

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**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 #16 Section 11.4

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**Using Pascal’s Triangle**

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

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

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