Adding, subtracting, multiplying, dividing of rational expressions polynomials Examples

1. Find 1x+1+xx-1 $(1)/(x+1) + (x)/(x-1)$

Solution:

=1x+1+xx-1 $=1/(x+1)+x/(x-1)$

=1×(x-1)+x×(x+1)(x+1)(x-1) $=(1 xx (x-1)+x xx (x+1))/((x+1)(x-1))$

=(x-1)+(x2+x)(x+1)(x-1) $=((x-1)+(x^2+x))/((x+1)(x-1))$

=2x-1+x2(x+1)(x-1) $=(2x-1+x^2)/((x+1)(x-1))$

2. Find 1x+1-xx-1 $(1)/(x+1) - (x)/(x-1)$

Solution:

=1x+1-xx-1 $=1/(x+1)-x/(x-1)$

=1×(x-1)-x×(x+1)(x+1)(x-1) $=(1 xx (x-1)-x xx (x+1))/((x+1)(x-1))$

=(x-1)-(x2+x)(x+1)(x-1) $=((x-1)-(x^2+x))/((x+1)(x-1))$

=-1-x2(x+1)(x-1) $=(-1-x^2)/((x+1)(x-1))$

3. Find x-1x+1⋅xx-1 $(x-1)/(x+1) * (x)/(x-1)$

Solution:

=x-1x+1×xx-1 $=(x-1)/(x+1)×x/(x-1)$

=xx+1 $=x/(x+1)$

4. Find 1x-1/xx-1 $(1)/(x-1) / (x)/(x-1)$

Solution:

=1x-1/(xx-1) $=1/(x-1)/(x/(x-1))$

=1x-1×x-1x $=1/(x-1)×(x-1)/x$

=1x $=1/x$

5. Find 1x-1+xx-1-1x+1 $(1)/(x-1) + (x)/(x-1) - (1)/(x+1)$

Solution:

=1x-1+xx-1-1x+1 $=1/(x-1)+x/(x-1)-1/(x+1)$

=1×(x+1)+x×(x+1)-1×(x-1)(x-1)(x+1) $=(1 xx (x+1)+x xx (x+1)-1 xx (x-1))/((x-1)(x+1))$

=(x+1)+(x2+x)-(x-1)(x-1)(x+1) $=((x+1)+(x^2+x)-(x-1))/((x-1)(x+1))$

=x+2+x2(x-1)(x+1) $=(x+2+x^2)/((x-1)(x+1))$

6. Find 1x-1/xx-1⋅1x+1 $(1)/(x-1) / (x)/(x-1) * (1)/(x+1)$

Solution:

=1x-1/(xx-1)×1x+1 $=1/(x-1)/(x/(x-1))×1/(x+1)$

=1x-1×x-1x×1x+1 $=1/(x-1)×(x-1)/x×1/(x+1)$

=1x2+x $=1/(x^2+x)$

7. Find x-1x+1-x+1x-1+4xx2-1 $(x-1)/(x+1) - (x+1)/(x-1) + (4x)/(x^2-1)$

Solution:

=x-1x+1-x+1x-1+4xx2-1 $=(x-1)/(x+1)-(x+1)/(x-1)+(4x)/(x^2-1)$

=x-1x+1-x+1x-1+4x(x-1)(x+1) $=(x-1)/(x+1)-(x+1)/(x-1)+(4x)/((x-1)(x+1))$

=(x-1)×(x-1)-(x+1)×(x+1)+4x(x+1)(x-1) $=((x-1) xx (x-1)-(x+1) xx (x+1)+4x)/((x+1)(x-1))$

=(x2-2x+1)-(x2+2x+1)+4x(x+1)(x-1) $=((x^2-2x+1)-(x^2+2x+1)+4x)/((x+1)(x-1))$

=0(x+1)(x-1) $=0/((x+1)(x-1))$

=0 $=0$

8. Find x+2x2-4x+3-x+3x2-3x+2+x+1x2-5x+6 $(x+2)/(x^2-4x+3) - (x+3)/(x^2-3x+2) + (x+1)/(x^2-5x+6)$

Solution:

=x+2x2-4x+3-x+3x2-3x+2+x+1x2-5x+6 $=(x+2)/(x^2-4x+3)-(x+3)/(x^2-3x+2)+(x+1)/(x^2-5x+6)$

=x+2(x-1)(x-3)-x+3(x-1)(x-2)+x+1(x-2)(x-3) $=(x+2)/((x-1)(x-3))-(x+3)/((x-1)(x-2))+(x+1)/((x-2)(x-3))$

=(x+2)×(x-2)-(x+3)×(x-3)+(x+1)×(x-1)(x-3)(x-2)(x-1) $=((x+2) xx (x-2)-(x+3) xx (x-3)+(x+1) xx (x-1))/((x-3)(x-2)(x-1))$

=(x2-4)-(x2-9)+(x2-1)(x-3)(x-2)(x-1) $=((x^2-4)-(x^2-9)+(x^2-1))/((x-3)(x-2)(x-1))$

=4+x2(x-3)(x-2)(x-1) $=(4+x^2)/((x-3)(x-2)(x-1))$

9. Find x2+5x+4x2-4⋅x2-5x+6x2-16/x2-2x-3x2-2x-8 $(x^2+5x+4)/(x^2-4) * (x^2-5x+6)/(x^2-16) / (x^2-2x-3)/(x^2-2x-8)$

Solution:

=x2+5x+4x2-4×x2-5x+6x2-16/(x2-2x-3x2-2x-8) $=(x^2+5x+4)/(x^2-4)×(x^2-5x+6)/(x^2-16)/((x^2-2x-3)/(x^2-2x-8))$

=(x+1)(x+4)(x-2)(x+2)×(x-2)(x-3)(x-4)(x+4)/((x+1)(x-3)(x+2)(x-4)) $=((x+1)(x+4))/((x-2)(x+2))×((x-2)(x-3))/((x-4)(x+4))/(((x+1)(x-3))/((x+2)(x-4)))$

=(x+1)(x+4)(x-2)(x+2)×(x-2)(x-3)(x-4)(x+4)×(x+2)(x-4)(x+1)(x-3) $=((x+1)(x+4))/((x-2)(x+2))×((x-2)(x-3))/((x-4)(x+4))×((x+2)(x-4))/((x+1)(x-3))$

=1 $=1$

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