How could I find the coefficient of the term with the variable b^5w^4 in the polynomial:(1/7)(b+w)^9 - (1/3)*(b+w)*(b^2-w^2)^4 + (b+w)^5*(b^4-w^4)

justaguide | College Teacher | (Level 2) Distinguished Educator

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The expression we have is (1/7)*(b+w)^9 - (1/3)*(b+w)(b^2-w^2)^4 + (b+w)^5(b^4-w^4)

Simplify the expression:

(1/7)*(b+w)^9

This involves the use of the binomial theorem for a large value of 9. I have only given the final result.

=>(w^9 + 9*b*w^8 + 36*b^2*w^7 + 84*b^3*w^6 + 126*b^4*w^5  + 126*b^5*w^4 +84*b^6*w^3 + 36*b^7*w^2 + 9*b^8*w+b^9)/7

Similarly - (1/3)*(b+w)(b^2-w^2)^4

=> -(w^9 + b*w^8 - 4*b^2*w^7 - 4*b^3*w^6 + 6*b^4*w^5 + 6*b^5*w^4 - 4*b^6*w^3 - 4*b^7*w^2 + b^8*w+b^9)/3

(b+w)^5(b^4-w^4)

=> -w^9 - 5*b*w^8 - 10*b^2*w^7 - 10*b^3*w^6 - 4*b^4*w^5 + 4*b^5*w^4 + 10*b^6*w^3 + 10*b^7*w^2 + 5*b^8*w + b^9

Adding up the three expansions and equating the denominator we get:

-(25*w^9 + 85*b*w^8 + 74*b^2*w^7 - 70*b^3*w^6 - 252*b^4*w^5 - 420*b^5*w^4 - 490*b^6*w^3 - 346*b^7*w^2 - 125*b^8*w - 17*b^9)/21

The coefficient of b^5*w^4 is 20