Coefficient of x^35 in Binomial Theorem Expansion

In summary, the question is asking for the coefficient of x^35 in the expansion of (a+b)^15 using the Binomial Theorem. The binomial coefficients are affected by the higher powers inside the binomial, and the exponents can be rewritten as (x^5)^7 and (y^6)^8. The binomial coefficient for the a^7b^8 term can be found by considering the number of -3s that are picked up.
  • #1
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Homework Statement


Find the coefficient of [PLAIN]http://webwork2.math.utah.edu/webwork2_files/tmp/equations/73/3e29a3b979c709dbb6c609c5a6ce891.png in the expansion of [PLAIN]http://webwork2.math.utah.edu/webwork2_files/tmp/equations/63/dcb58790e8122dce61b830977294091.png

Homework Equations


The Binomial Theorem

The Attempt at a Solution


This one is stumping me. I guess because in all of my previous problems, we didn't have any binomials which had parts raised to a higher power like (-3x)^5

Anyway, would my r in my nCr portion of the theorem be 15C34? Seeing as I'm attempting to find the term which has x^35th?

Typing that out, I think my question is actually, does the higher powers inside of the binomial have any effect on the theorem?

My attempt if this is true:

15C34(((-3a^5)^-19)-(3y^6)^34)

Typing that out looks horribly wrong. Some help please?
 
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  • #2
Hrm, can't really understand what your attempt is saying.
Just picture the exponents differently:

[tex]x^{35} = (x^5)^7[/tex]

[tex]y^{48} = (y^6)^8[/tex]

Now you need to find the binomial coefficient for the [tex]a^7b^8[/tex] term in the expansion of[tex](a+b)^{15}[/tex]. Don't forget you will pick up some -3s. How many?
 

1. What is the Coefficient of x^35 in Binomial Theorem Expansion?

The coefficient of x^35 in Binomial Theorem Expansion is the number that is multiplied by the term x^35. It is calculated using the formula (n choose k) * a^(n-k) * b^k, where n is the power of the binomial, k is the power of x in the term, and a and b are the coefficients of the binomial terms.

2. How do you find the Coefficient of x^35 in Binomial Theorem Expansion?

To find the coefficient of x^35 in Binomial Theorem Expansion, you can use the formula (n choose k) * a^(n-k) * b^k, where n is the power of the binomial, k is the power of x in the term, and a and b are the coefficients of the binomial terms. You can also use Pascal's Triangle or expand the binomial using the binomial theorem and then identify the term with x^35.

3. Can the Coefficient of x^35 in Binomial Theorem Expansion be negative?

Yes, the coefficient of x^35 in Binomial Theorem Expansion can be negative. This depends on the coefficients of the binomial terms and the value of n and k in the formula (n choose k) * a^(n-k) * b^k. It is important to note that the coefficient can only be negative if the term with x^35 has a negative coefficient.

4. What is the significance of the Coefficient of x^35 in Binomial Theorem Expansion?

The coefficient of x^35 in Binomial Theorem Expansion is significant because it tells us the number of ways to choose k elements from a set of n elements and the power of x in the term. This is useful in many mathematical and scientific fields, such as probability, statistics, and combinatorics.

5. Can the Coefficient of x^35 in Binomial Theorem Expansion be a decimal or fraction?

Yes, the coefficient of x^35 in Binomial Theorem Expansion can be a decimal or fraction. This depends on the values of n and k in the formula (n choose k) * a^(n-k) * b^k. The coefficient can only be a decimal or fraction if the term with x^35 has a decimal or fraction as its coefficient.

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