Binomial q (1 Viewer)

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Q14b pls

 

[ExtremelyBoredUser]

Set up the Binomial Expansion and find the coeffs - should be straightforward. I got;

Coefficient of x^8 is nc8 * 1^(n-8) * 3^8
Coefficient of x^10 is nc10 * 1^(n-10) * 3^10

1 is ignored since any value of n will produce 1 regardless

These coefficients are in a ratio of 1:2 therefore;

2(nc8 * 3^8) = nc10 * 3^10

Simplifying we get;

nc8 = nc10 * 9/2
n!/(n-8)!8! = n!/(n-10)!10! * 9/2
1/(n-8)(n-9)8! = 1/10! * 9/2
1/(n-8)(n-9) = 1/90 * 9/2 = 1/20

taking reciprocal:

(n-8)(n-9) = 20
n^2 - 17n + 72 - 20 = 0
n^2 - 17n + 52 = 0
n = 13, 4
n =/ 4 therefore n = 13

4 would be incorrect because if you substitute into the equation it would create a negative factorial which is not computable. Therefore 13 is the correct answer. Someone check this since I always make typos somewhere when i type.

 

[fnsndgfg]

These coefficients are in a ratio of 1:2 therefore;

2(nc8 * 3^8) = nc10 * 3^10

Hi, so I also struggle with polynomials, when you did this how did you know the coefficients were equal? Does it have something to do with pascals triangle?

 

[ExtremelyBoredUser]

Hi, so I also struggle with polynomials, when you did this how did you know the coefficients were equal? Does it have something to do with pascals triangle?

It stated that the coefficients were in a ratio of 1:2. This meant that the x^8 coeff was double of x^10 coeff. Ratios by nature demonstrate an equality between two variables, it's just another way of expressing a fraction; that's how I interpret it.

To make it more clear, you can consider ratios as fractions.

So

x^8 coeff/x^10 coeff = 1/2

This mathematical statement is the same as the written statement x^8 and x^10 are in a ratio of 1:2.

Hence when you rearrange you would get

2 * (x^8 coeff) = 1 * (x^10 coeff)

 

[fnsndgfg]

It stated that the coefficients were in a ratio of 1:2. This meant that the x^8 coeff was double of x^10 coeff. Ratios by nature demonstrate an equality between two variables, it's just another way of expressing a fraction; that's how I interpret it.

To make it more clear, you can consider ratios as fractions.

So

x^8 coeff/x^10 coeff = 1/2

This mathematical statement is the same as the written statement x^8 and x^10 are in a ratio of 1:2.

Hence when you rearrange you would get

2 * (x^8 coeff) = 1 * (x^10 coeff)

Ahh that makes sense, I was being a bit of an idiot