Expand Using Binomial Theorem: (1-y^2)^5

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In summary, the conversation discusses the use of the binomial theorem to expand and simplify the expression (1-y^2)^5. The binomial theorem is used to find the coefficients of each term, which can also be interpreted as the number of ways to choose a certain number of variables in each term. The resulting simplified expression is 1 - 5y^2 + 10y^4 - 10y^6 - 5y^8 - y^10. The conversation also touches on the symmetry of binomial coefficients and how they form Pascal's Triangle.
  • #1
Styx
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Use the binomial theorem to expand each of the following. Simpify your answers

(1-y^2)^5

Let a = 1 and b = -(y^2)
Then using binomial theorem, you have:

(a+b)^5 = C(5,0)a^5 + C(5,1)a^4 b + C(5,2)a^3 b^2 + C(5,3)a^2 b^3

+ C(5,4)a b^4 + C(5,5)b^5

Substitute a = 1 and b = -(y^2)

(1-y^2)^5 = 1(1)^5 + 5(1)^4 (-(y^2)) + 10(1)^3 (-(y^2))^2

+ 10(1)^2 (-(y^2))^3 + 5(1)(-(y^2))^4 + 1(-(y^2))^5

= 1 + 5(-(y^2)) + 10y^4 + 10(-(y^6)) + 5y^8 + (-(y^10))

= 1 -5y^2 + 10y^4 -10y^6 -5y^8 -y^10

Does that look right? This is my first go at using binomial theorem...
 
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  • #2
It looks good to me. Your binomial coefficients (the blah "choose" blah factors) make sense if you interpret them as the number of ways of choosing the number of b's out of 5 in each term. You could also formulate it by talking about the number of ways of choosing the number of a's out of 5 in each term. You'd get the same answer, because the binomial coefficients are always symmetric (for example, 5 choose 0 equals 5 choose 5, because the number of ways of choosing all of the elements in a set is the same as the number of ways of choosing none of them: 1. Another example: 5 choose 4 equals 5 choose 1 because the number of ways of choosing 4 elements out of 5 is the same as the number of ways of excluding one of them: 5). This explains why the binomial coefficients build up Pascal's Triangle (all of the rows in the triangle are symmetric and bounded by 1's). I hope this offers further insight.
 
  • #3
thanks cepheid
 

What is the binomial theorem?

The binomial theorem is a mathematical formula that allows for the expansion of expressions in the form of (a + b)^n, where a and b are constants and n is a positive integer. It is used to simplify and solve problems involving binomial expressions.

How is the binomial theorem applied to (1-y^2)^5?

To expand (1-y^2)^5, we can use the binomial theorem by substituting a=1, b=-y^2, and n=5 into the formula (a + b)^n. This will give us the expanded expression of 1 - 5y^2 + 10y^4 - 10y^6 + 5y^8 - y^10.

What is the significance of the exponent in (1-y^2)^5?

The exponent in (1-y^2)^5 represents the number of times the expression is multiplied by itself. In this case, the expression is being multiplied 5 times, which is why the exponent is 5.

Can the binomial theorem be used for expressions with negative exponents?

Yes, the binomial theorem can be used for expressions with negative exponents. However, the expression must be rewritten in the form of (1+x)^n, where x is the negative exponent. For example, (1-y^-2)^5 can be rewritten as (1+y^2)^5, which can then be expanded using the binomial theorem.

What are some real-life applications of the binomial theorem?

The binomial theorem has many real-life applications in fields such as physics, engineering, and finance. It can be used to solve problems involving probability, geometric series, and compound interest. It is also used in the expansion of polynomial functions and in the development of mathematical models.

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