Asked by SimPug
                   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
            
        (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
Answers
                    Answered by
            Steve
            
    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
    
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
                    Answered by
            scott
            
    google "polynomial multiplier calculator"
use the easycalculation application
    
use the easycalculation application
                    Answered by
            Reiny
            
    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
    
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
                    Answered by
            scott
            
    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
    
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
                    Answered by
            Reiny
            
    Thanks, why didn't I see that ?
    
                    Answered by
            Shivansh Soni
            
    378 is the correct ans. 
    
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