EXAMPLES ON ADDING AND SUBTRACTING RATIONAL EXPRESSIONS

Problem 1 :

2z1-2z+3z2z+1-34z2-1

Solution:

2z1-2z+3z2z+1-34z2-1By comparing 4z2-1 with the algebraic identitya2-b2=(a+b)(a-b)4z2-1=(2z+1)(2z-1)=2z1-2z+3z2z+1-3(2z+1)(2z-1)LCM=(1-2z)(2z+1)=2z1-2z×2z+12z+1+3z2z+1×1-2z1-2z+3(2z+1)(1-2z)=2z(2z+1)+3z(1-2z)+3(1-2z)(2z+1)=4z2+2z+3z-6z2+3(1-2z)(2z+1)=-2z2+5z+3(1-2z)(2z+1)=-(1+2z)(z-3)(1-2z)(2z+1)=-(z-3)1-2z=z-32z-1

Problem 2 :

2rr2-s2+1r+s-1r-s

Solution:

2rr2-s2+1r+s-1r-sBy comparing r2-s2 with the algebraic identitya2-b2=(a+b)(a-b)r2-s2=(r+s)(r-s)=2r(r+s)(r-s)+1r+s-1r-sLCM=(r+s)(r-s)=2r(r+s)(r-s)+1r+s×r-sr-s-1r-s×r+sr+s=2r+r-s-r-s(r+s)(r-s)=2r-2s(r+s)(r-s)=2(r-s)(r+s)(r-s)=2r+s

Problem 3 :

2x-3x2+3x+2+3x-1x2+5x+6

Solution:

2x-3x2+3x+2+3x-1x2+5x+6x2+3x+2=(x+1)(x+2)x2+5x+6=(x+2)(x+3)=2x-3(x+1)(x+2)+3x-1(x+2)(x+3)LCM=(x+1)(x+2)(x+3)=2x-3(x+1)(x+2)×x+3x+3+3x-1(x+2)(x+3)×x+1x+1=(2x-3)(x+3)+(3x-1)(x+1)(x+1)(x+2)(x+3)=2x2+6x-3x-9+3x2+3x-x-1(x+1)(x+2)(x+3)=5x2+5x-10(x+1)(x+2)(x+3)=5x2+x-2(x+1)(x+2)(x+3)=5(x-1)(x+2)(x+1)(x+2)(x+3)=5(x-1)(x+1)(x+3)

Problem 4 :

x+2x2-4x+3+4x+5x2+4x-5

Solution:

x+2x2-4x+3+4x+5x2+4x-5x2-4x+3=(x-1)(x-3)x2+4x-5=(x-1)(x+5)=x+2(x-1)(x-3)+4x+5(x-1)(x+5)LCM=(x-1)(x-3)(x+5)=x+2(x-1)(x-3)×x+5x+5+4x+5(x-1)(x+5)×x-3x-3=(x+2)(x+5)+(4x+5)(x-3)(x-1)(x-3)(x+5)=x2+7x+10+4x2-7x-15(x-1)(x-3)(x+5)=5x2-5(x-1)(x-3)(x+5)=5x2-1(x-1)(x-3)(x+5)=5(x+1)(x-1)(x-1)(x-3)(x+5)=5(x+1)(x-3)(x+5)=5x+5x2+2x-15

Problem 5 :

2x+7x2-2x-3-3x-2x2+6x+5

Solution:

2x+7x2-2x-3-3x-2x2+6x+5x2-2x-3=(x+1)(x-3)x2+6x+5=(x+1)(x+5)=2x+7(x+1)(x-3)-3x-2(x+1)(x+5)LCM=(x+1)(x-3)(x+5)=2x+7(x+1)(x-3)×x+5x+5-3x-2(x+1)(x+5)×x-3x-3=(2x+7)(x+5)-(3x-2)(x-3)(x+1)(x-3)(x+5)=2x2+17x+35-3x2+11x-6(x+1)(x-3)(x+5)=-x2+28x+29(x+1)(x-3)(x+5)=-(x+1)(x-29)(x+1)(x-3)(x+5)=-(x-29)(x-3)(x+5)

Problem 6 :

3x-8x2+6x+8+2x-3x2+3x+2

Solution:

3x-8x2+6x+8+2x-3x2+3x+2x2+6x+8=(x+2)(x+4)x2+3x+2=(x+1)(x+2)=3x-8(x+2)(x+4)+2x-3(x+1)(x+2)LCM=(x+1)(x+2)(x+4)=3x-8(x+2)(x+4)×x+1x+1+2x-3(x+1)(x+2)×x+4x+4=(3x-8)(x+1)+(2x-3)(x+4)(x+1)(x+2)(x+4)=3x2-5x-8+2x2+5x-12(x+1)(x+2)(x+4)=5x2-20(x+1)(x+2)(x+4)=5x2-4(x+1)(x+2)(x+4)=5(x+2)(x-2)(x+1)(x+2)(x+4)=5(x-2)(x+1)(x+4)=5x-10x2+5x+4

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