Question

In the expansion of
(1 + x + x^2 + ...+ x^27) (1 + x + x^2 + ... + x^14) ^2

What is the coefficient of x^28?

A) 195 B)196 C)224 D)378

Me and my sister are both trying to solve this for school, but keep getting stuck and are confused on it. If you could help us we will be very thankful

Answer 1

There are 28 terms on the left, of degree k, where k=0..27
Each of those is paired with a term on the right of degree 28-k

(1+x+x^2+...+x^14)^2 = 1+2x+3x^2+...+14x^13+15x^14+14x^15+...+2x^27+x^28

Play around with that.
the sum of numbers from 1 to n is n(n+1)/2, but it's not quite just that simple.
You should come up with 224

Answer 2

google "polynomial multiplier calculator"

use the easycalculation application

Answer 3

I hope you weren't trying to expand all of this.
let's look at the last part first, by looking at a some patterns
(1+x)^2 = 1 + 2x + x^2 , 3 terms, coefficients run 1,2,1
(1+x+x^2) = 1 + 2x + 3x^2 + 2x^3 + x^4, 5 terms, coefficients run 1,2,3,2,1
(1+x+x^2+x^3)^2 = 1 + 2x + 3x^2 + 4x^3 + 3x^4 + 2x^5 + x^6 , 7terms, coefficients run 1,2,3,4,3,2,1
(1+x+x^2+x^3+...+x^13+x^14)
= 1+2x+3x^2+4x^3+5x^4+6x^5+...+13x^14+14x^15+13x^16+...+2x^27+x^28 , ---> 27 terms, coefficients run 1 2 3 4 5 6 7 8 9 10 11 12 13 14 13 12 11 10 9 8 7 6 5 4 3 2 1

now if we multiply this by (1 + x + x^2 + ...+ x^27) , where can the term containing x^28 come from?
that would be:
x*2x^27 , x^2*3x^26 , x^3*4x^25, x^4*5x^24, .... , x^13*14x^15+x^14*13x^14 .... x^26*3x^2+x^27*2x

so our coefficients add up to :
2+3+4+5+6+...+13+14+13+...+3+2
= 90 + 14 + 90
= 194

check my arithmetic, I can't find any errors

Answer 4

Reiny

following your pattern; if x^n is the largest term , then there are 2n+1 terms

the (increasing) term coefficients are one greater than their corresponding exponent

you're missing some terms/coefficients in the squaring of the smaller polynomial

the easycalculation application agrees with Steve

Answer 5

Thanks, why didn't I see that ?

Answer 6

378 is the correct ans.