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
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
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
google "polynomial multiplier calculator"
use the easycalculation application
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
Thanks, why didn't I see that ?
378 is the correct ans.
Of course! I'll be happy to help you and your sister solve this problem.
To find the coefficient of x^28 in the given expression, we need to sum up all the terms that contribute to x^28.
Let's break down the expression step by step:
(1 + x + x^2 + ... + x^27) (1 + x + x^2 + ... + x^14)^2
We can rewrite the terms in each parentheses using the formula for the sum of a geometric series:
For the first term in the parentheses, we have a geometric series with a common ratio of x and 28 terms:
1 + x + x^2 + ... + x^27 = (1 - x^28) / (1 - x)
For the second term in the parentheses, we have a geometric series with a common ratio of x and 15 terms:
1 + x + x^2 + ... + x^14 = (1 - x^15) / (1 - x)
Now, let's substitute these expressions back into the original equation:
[(1 - x^28) / (1 - x)] * [(1 - x^15) / (1 - x)]^2
Next, we can simplify this expression by multiplying the numerators and denominators:
(1 - x^28)(1 - x^15)^2 / (1 - x)^3
Expanding the numerator:
(1 - 2x^15 + x^30)(1 - x^28) / (1 - x)^3
Now, we distribute and combine like terms:
(1 - 2x^15 + x^30 - x^28 + 2x^43 - x^58) / (1 - x)^3
To find the coefficient of x^28, we need to look at the terms that have an x^28 factor. In this case, the only term with x^28 is -x^28.
Therefore, the coefficient of x^28 is -1.
So, the answer is not given in the options A, B, C, or D. It seems there might be an error in the given options. You may want to notify your teacher about this discrepancy.
I hope this explanation helps! If you have any further questions, please feel free to ask.