Algebra Help: find numerical coefficient of x^27 in...

[osoriopinto5]

I'm not sure if this is Algebra, but I need to know if someone can give me the first step in solving this problem.

Find numerical coefficient of x^27 in complete expansion of

(x^2 + 5x+ 2)^5 times (x^2 - 7x + 3)^9

 

[galactus]

Actually, it's very easy to find. Take the middle coefficient in each expansion, 5 and -7.

Multiply the exponent of the first expansion,5, by the coffiecient of the middle term, 5. 5*5=25.

Do the same for the other expansion.

Multiply the exponent of the expansion, 9, by the coefficient of the middle term, -7. 9*(-7)=-63

Now, add them -63+25=-38

 

[osoriopinto5]

will that always work or in only certain cases?

 

[osoriopinto5]

oh, and also why does that only work for x^27.

 

[galactus]

It works for this particular type of problem. Try some others. You can find patterns and see how to solve these without expanding. If you had to expand these, it would be a booger

 

[osoriopinto5]

thanks

that really helps