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Simplify the following algebraic expressions: (1/(x-y)) - (1/(x+y)) + (2x/(x2-y2)) AND ((x+2)/(1+x+x2)) - ((x-2)/(1-x+x2)) - (2x2/(1+x2+x4)).

Mathematics

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We need to simplify the expression 1x−y−1x+y+2xx2−y2\frac{1}{x-y} - \frac{1}{x+y} + \frac{2x}{x^2-y^2}.

  1. Combine the first two fractions by finding a common denominator, which is (x−y)(x+y)=x2−y2(x-y)(x+y) = x^2-y^2.

    1x−y−1x+y=1⋅(x+y)(x−y)(x+y)−1⋅(x−y)(x+y)(x−y)\frac{1}{x-y} - \frac{1}{x+y} = \frac{1 \cdot (x+y)}{(x-y)(x+y)} - \frac{1 \cdot (x-y)}{(x+y)(x-y)} =x+y−(x−y)x2−y2= \frac{x+y - (x-y)}{x^2-y^2} =x+y−x+yx2−y2= \frac{x+y-x+y}{x^2-y^2} =2yx2−y2= \frac{2y}{x^2-y^2}

  2. Add the result from Step 1 to the third fraction.

    2yx2−y2+2xx2−y2=2y+2xx2−y2\frac{2y}{x^2-y^2} + \frac{2x}{x^2-y^2} = \frac{2y+2x}{x^2-y^2}

  3. Factor the numerator and the denominator.

    2(y+x)(x−y)(x+y)\frac{2(y+x)}{(x-y)(x+y)} =2(x+y)(x−y)(x+y)= \frac{2(x+y)}{(x-y)(x+y)}

  4. Cancel out the common term (x+y)(x+y).

    =2x−y= \frac{2}{x-y}

    The simplified expression is 2x−y\boxed{\frac{2}{x-y}}.

    :
    We need to simplify the expression x+21+x+x2−x−21−x+x2−2x21+x2+x4\frac{x+2}{1+x+x^2} - \frac{x-2}{1-x+x^2} - \frac{2x^2}{1+x^2+x^4}.

  5. Find a common denominator for the first two fractions.

    The product of their denominators is (1+x+x2)(1−x+x2)=(1+x2+x)(1+x2−x)=(1+x2)2−x2=1+2x2+x4−x2=1+x2+x4(1+x+x^2)(1-x+x^2) = (1+x^2+x)(1+x^2-x) = (1+x^2)^2 - x^2 = 1+2x^2+x^4-x^2 = 1+x^2+x^4.

    (x+2)(1−x+x2)−(x−2)(1+x+x2)(1+x+x2)(1−x+x2)\frac{(x+2)(1-x+x^2) - (x-2)(1+x+x^2)}{(1+x+x^2)(1-x+x^2)} =(x−x2+x3+2−2x+2x2)−(x+x2+x3−2−2x−2x2)1+x2+x4= \frac{(x-x^2+x^3+2-2x+2x^2) - (x+x^2+x^3-2-2x-2x^2)}{1+x^2+x^4} =(x3+x2−x+2)−(x3−x2−x−2)1+x2+x4= \frac{(x^3+x^2-x+2) - (x^3-x^2-x-2)}{1+x^2+x^4}

  6. Simplify the numerator.

    x3+x2−x+2−x3+x2+x+21+x2+x4\frac{x^3+x^2-x+2 - x^3+x^2+x+2}{1+x^2+x^4} =(x3−x3)+(x2+x2)+(−x+x)+(2+2)1+x2+x4= \frac{(x^3-x^3) + (x^2+x^2) + (-x+x) + (2+2)}{1+x^2+x^4} =2x2+41+x2+x4= \frac{2x^2+4}{1+x^2+x^4}

  7. Subtract the third fraction from the result of Step 2.

    2x2+41+x2+x4−2x21+x2+x4\frac{2x^2+4}{1+x^2+x^4} - \frac{2x^2}{1+x^2+x^4} =2x2+4−2x21+x2+x4= \frac{2x^2+4-2x^2}{1+x^2+x^4} =41+x2+x4= \frac{4}{1+x^2+x^4}

    The simplified expression is 41+x2+x4\boxed{\frac{4}{1+x^2+x^4}}.

    :
    We need to simplify the expression 1b+c+b+cb2−c2\frac{1}{b+c} + \frac{b+c}{b^2-c^2}.

  8. Simplify the second term by factoring the denominator b2−c2=(b−c)(b+c)b^2-c^2 = (b-c)(b+c).

    b+cb2−c2=b+c(b−c)(b+c)\frac{b+c}{b^2-c^2} = \frac{b+c}{(b-c)(b+c)} =1b−c= \frac{1}{b-c}

  9. Add the first term to the simplified second term.

    1b+c+1b−c\frac{1}{b+c} + \frac{1}{b-c}

  10. Find a common denominator, which is (b+c)(b−c)=b2−c2(b+c)(b-c) = b^2-c^2.

    =1⋅(b−c)(b+c)(b−c)+1⋅(b+c)(b−c)(b+c)= \frac{1 \cdot (b-c)}{(b+c)(b-c)} + \frac{1 \cdot (b+c)}{(b-c)(b+c)} =b−c+b+cb2−c2= \frac{b-c+b+c}{b^2-c^2}

  11. Simplify the numerator.

    =2bb2−c2= \frac{2b}{b^2-c^2}

    The simplified expression is 2bb2−c2\boxed{\frac{2b}{b^2-c^2}}.

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