Simplify the following algebraic expressions: 1. 1/(1+x2) + 2/(1+x2) − 4/(1-x4) 2. (a-b)/(a2-ab+b2)…

Question

Simplify the following algebraic expressions: 1. 1/(1+x2) + 2/(1+x2) − 4/(1-x4) 2. (a-b)/(a2-ab+b2) + (a+b)/(a2+ab+b2) − 2b3/(a4+a2b2+b4)

ScanSolve AI Tutor · Accepted Answer

Here are the solutions to the problems:

1. Simplify: $\frac{1}{1+x^2} + \frac{2}{1+x^2} - \frac{4}{1-x^4}$

Combine the first two terms since they have the same denominator.
$\frac{1+2}{1+x^2} - \frac{4}{1-x^4} = \frac{3}{1+x^2} - \frac{4}{1-x^4}$
Factor the denominator of the second term using the difference of squares formula, $A^2-B^2 = (A-B)(A+B)$ .
Here, $1-x^4 = (1-x^2)(1+x^2)$ .
$\frac{3}{1+x^2} - \frac{4}{(1-x^2)(1+x^2)}$
Find the common denominator, which is $(1-x^2)(1+x^2)$ .
Multiply the numerator and denominator of the first term by $(1-x^2)$ .
$\frac{3(1-x^2)}{(1-x^2)(1+x^2)} - \frac{4}{(1-x^2)(1+x^2)}$
Combine the numerators and simplify.
$\frac{3(1-x^2) - 4}{(1-x^2)(1+x^2)} = \frac{3-3x^2-4}{1-x^4} = \frac{-1-3x^2}{1-x^4}$ $= \boxed{\frac{-(1+3x^2)}{1-x^4}}$
2. Simplify: $\frac{a-b}{a^2-ab+b^2} + \frac{a+b}{a^2+ab+b^2} - \frac{2b^3}{a^4+a

Simplify the following algebraic expressions: 1. 1/(1+x2) + 2/(1+x2) - 4/(1-x4) 2. (a-b)/(a2-ab+b2) + (a+b)/(a2+ab+b2) - 2b3/(a4+a2b2+b4)