ADDING AND SUBTRACTING RATIONAL EXPRESSIONS EXAMPLES

Problem 1 :

2xx2-1-4x2+2x-3

Solution:

2xx2-1-4x2+2x-3By comparing x2-1 with the algebraic identitya2-b2=(a+b)(a-b)x2-1=(x+1)(x-1)x2+2x-3=(x-1)(x+3)=2x(x+1)(x-1)-4(x-1)(x+3)LCM=(x+1)(x-1)(x+3)=2x(x+1)(x-1)×(x+3)(x+3)-4(x-1)(x+3)×(x+1)(x+1)=2x(x+3)-4(x+1)(x+1)(x-1)(x+3)=2x2+6x-4x-4(x+1)(x-1)(x+3)=2x2+2x-4(x+1)(x-1)(x+3)=2x2+x-2(x+1)(x-1)(x+3)=2(x-1)(x+2)(x+1)(x-1)(x+3)=2(x+2)(x+1)(x+3)=2x+4x2+4x+3

Problem 2 :

x-1x2+3x+2+x+5x2+4x+3

Solution:

x-1x2+3x+2+x+5x2+4x+3x2+3x+2=(x+1)(x+2)x2+4x+3=(x+1)(x+3)=x-1(x+1)(x+2)+x+5(x+1)(x+3)LCM=(x+1)(x+2)(x+3)=(x-1)(x+1)(x+2)×(x+3)(x+3)+x+5(x+1)(x+3)×(x+2)(x+2)=(x-1)(x+3)+(x+5)(x+2)(x+1)(x+2)(x+3)=x2+2x-3+x2+7x+10(x+1)(x+2)(x+3)=2x2+9x+7(x+1)(x+2)(x+3)=(x+1)(2x+7)(x+1)(x+2)(x+3)=2x+7(x+2)(x+3)=2x+7x2+5x+6

Problem 3 :

x+1x2-2x-35+x+6x2+7x+10

Solution:

x+1x2-2x-35+x+6x2+7x+10x2-2x-35=(x+5)(x-7)x2+7x+10=(x+5)(x+2)=x+1(x+5)(x-7)+x+6(x+5)(x+2)LCM=(x+2)(x+5)(x-7)=x+1(x+5)(x-7)×(x+2)(x+2)+x+6(x+5)(x+2)×(x-7)(x-7)=(x+1)(x+2)+(x+6)(x-7)(x+2)(x+5)(x-7)=x2+3x+2+x2-x-42(x+2)(x+5)(x-7)=2x2+2x-40(x+2)(x+5)(x-7)=2x2+x-20(x+2)(x+5)(x-7)=2(x-4)(x+5)(x+2)(x+5)(x-7)=2(x-4)(x+2)(x-7)=2x-8x2-5x-14

Problem 4 :

3x+23x+6+x4-x2

Solution:

3x+23x+6+x4-x23x+6=3(x+2)4-x2=22-x2By comparing 22-x2 with the algebraic identitya2-b2=(a+b)(a-b)22-x2=(2+x)(2-x)=3x+23(x+2)+x(2+x)(2-x)LCM=3(2+x)(2-x)=3x+23(x+2)×(2-x)(2-x)+x(2+x)(2-x)×33=(3x+2)(2-x)+3x3(x+2)(2-x)=6x-3x2+4-2x+3x3(x+2)(2-x)=-3x2+7x+43(x+2)(2-x)

Problem 5 :

4-a2a2-9-a-23-a

Solution:

4-a2a2-9-a-23-a4-a2=22-a2By comparing 22-a2 with the algebraic identitya2-b2=(a+b)(a-b)22-a2=(2+a)(2-a)a2-32=(a+3)(a-3)=(2+a)(2-a)(a+3)(a-3)-a-23-aLCM=(a+3)(a-3)=(2+a)(2-a)(a+3)(a-3)+a-23-a×(a+3)(a+3)=(2+a)(2-a)+(a-2)(a+3)(a+3)(a-3)=4-a2+a2-2a+3a-6a2-9=a-2a2-9

Problem 6 :

4yy2-1-2y-2y+1

Solution:

4yy2-1-2y-2y+1By comparing y2-1 with the algebraic identitya2-b2=(a+b)(a-b)y2-1=(y+1)(y-1)=4y(y+1)(y-1)-2y-2y+1LCM=y(y+1)(y-1)=4y(y+1)(y-1)×yy-2y×(y+1)(y-1)(y+1)(y-1)-2y+1×y(y-1)y(y-1)=4y2-2(y+1)(y-1)-2y(y-1)y(y+1)(y-1)=4y2-2y2-1-2y2+2yy(y+1)(y-1)=4y2-2y2+2-2y2+2yy(y+1)(y-1)=2+2yy(y+1)(y-1)=2(1+y)y(y+1)(y-1)=2y(y-1)=2y2-y

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