StudentAidUnit 5: Algebraic Fractions — Exercise 5 2
10th Class Mathematics · Unit 5: Algebraic Fractions
5.2.1.(i) Simplify: 3/(x-y) + 1/(y-x) (ii) Simplify: 4x/(x^2-3x+2) - 4/(1-x) - 5/(x-2) (iii) Simplify: (x+y)/(12x-6y) + (x-y)/(18x-9y) (iv) Simplify: 2/(x+2y) - (x-6y)/(x^2-4y^2) (v) Simplify: 1/(x+1) - 2/(x+2) - (2x+3)/(x^2+3x+2) (vi) Simplify: 5/(x^2+x-6) - 1/(2x^2-7x+6) (vii) Simplify: 2x/(x+y) - y/(x-y) - 2y^2/(x^2-y^2) (viii) Simplify: 7/(2x^2-x-6) - 8/(3x^2-4x-4) (ix) Simplify: (x+2)/(x^2-x-12) - x/(x^2+6x+9) (x) Simplify: 1/(x^2-4x+3) + 1/(x^2-3x+2) - 1/(x^2-5x+6) (xi) Simplify: x^3/(x-y) + y^3/(y-x)
(i)
3x-y+1y-x = 3x-y-1x-y
Result
(i)
= 2x-y
(ii) Common denominator
= 4x(x-1)(x-2) + 4x-1 - 5x-2
(ii)
= 4x+4(x-2)-5(x-1)(x-1)(x-2) = 3x-3(x-1)(x-2)
(ii)
= 3x-2
(iii) Factor denominators
= x+y6(2x-y) + x-y9(2x-y) = 3(x+y)+2(x-y)18(2x-y)
(iii)
= 5x+y18(2x-y)
(iv)
= 2x+2y - x-6y(x-2y)(x+2y) = 2(x-2y)-(x-6y)(x+2y)(x-2y)
(iv)
= x+2y(x+2y)(x-2y)
(iv)
= 1x-2y
(v)
= (x+2)-2(x+1)-(2x+3)(x+1)(x+2) = -3x-3(x+1)(x+2)
(v)
= -3x+2
(vi) Factor
= 5(x+3)(x-2) - 1(2x-3)(x-2) = 5(2x-3)-(x+3)(x+3)(x-2)(2x-3)
(vi)
= 9x-18(x+3)(x-2)(2x-3) = 9(x-2)(x+3)(x-2)(2x-3)
(vi)
= 9(x+3)(2x-3)
(vii)
= 2x(x-y)-y(x+y)-2y2(x+y)(x-y)
(vii)
= 2x2-3xy-3y2(x+y)(x-y)
(viii) Factor
= 7(2x+3)(x-2) - 8(3x+2)(x-2) = 7(3x+2)-8(2x+3)(2x+3)(x-2)(3x+2)
(viii)
= 5x-10(2x+3)(x-2)(3x+2) = 5(x-2)(2x+3)(x-2)(3x+2)
(viii)
= 5(2x+3)(3x+2)
(ix) Factor
= x+2(x-4)(x+3) - x(x+3)2 = (x+2)(x+3)-x(x-4)(x-4)(x+3)2
(ix)
= 9x+6(x-4)(x+3)2
(ix)
= 3(3x+2)(x-4)(x+3)2
(x) Factor
= 1(x-1)(x-3) + 1(x-1)(x-2) - 1(x-2)(x-3)
(x)
= (x-2)+(x-3)-(x-1)(x-1)(x-2)(x-3)
(x)
= x-4(x-1)(x-2)(x-3)
(xi)
= x3x-y - y3x-y = x3-y3x-y = (x-y)(x2+xy+y2)x-y
(xi)
= x2+xy+y2
5.2.2.Subtract 1/(x^2+2) from (2x^3+x^2+3)/(x^2+2)^2
Common denominator
2x3+x2+3(x2+2)2 - 1x2+2 = 2x3+x2+3-(x2+2)(x2+2)2
Result
= 2x3+1(x2+2)2