Question

If a and b are the roots of 11x^2-4x-2=0 then compute the product: (1+a+a^2+...)(1+b+b^2+...)

Answer 1

Answer:

11/5

Step-by-step explanation:

FirLet a and b be the roots of 11x^2-4x-2=0

Now$(1+a+a^2+\cdots)(1+b+b^2+\dcots)=\frac{1}{1-a}\cdot \frac{1}{1-b}\\=\frac{1}{(1-a)(1-b)}=\frac{1}{1-(a+b)+ab}$

Dividing the quadratic equation by 11 we have

$x^2-\frac{4}{11}x-\frac{2}{11}=0.$

Now the sum of the roots is the minus of the coefficient of x and the product of the roots is the constant. Therefore

$a+b = 4/11 \\ab = -2/11$

and we have

$\frac{1}{(1-a)(1-b)}=\frac{1}{1-\frac{4}{11}-\frac{2}{11}}\\=\frac{1}{(5/11)}=\frac{11}{5}$