गुणनखण्ड ज्ञात कीजिए :
(i) `x^(3)-2x(2)-x+2`
(ii) `x^(3)-3x^(2)-9x-5`
(iii) `x^(3)+13x^(2)+32x+20`
(iv) `2y^(3)+y^(2)-2y-1`

Explanation

(i) माना `p(x)=x^(3)-2x^(2)-x+2`
अचर पद =2
इसके गुणनखण्ड `pm1,pm2`
`:." "p(1)=1-2-1+2=0`
`rArrx-1,p(x)` का एक गुणनखण्ड है |
`:.x^(3)-2x^(2)-x+2`
`=x^(2)(x-1)-x^(2)-x+2`
`=x^(2)(x-1)-x(x-1)-2x+2`
`=x^(2)(x-1)-x(x-1)-2(x-1)`
`=(x-1)(x^(2)-x-2)`
`=(x-1)[x^(2)-2x+x-2]`
`=(x-1)[x(x-2)+1(x-2)]`
`=(-1)(x-2)(x+1)`
(ii) माना `p(x)=x^(3)-3x^(2)-9x-5`
यहाँ अचर पद `=5`
इसके गुणनखण्ड `=pm1,pm5`
`:." "p(-1)=(-1)^(3)-3(-1)^(2)-9(-1)-5`
`=-1-3+9-5=0`
`rArr" "x+1,p(x)` का एक गुणनखण्ड है ।
`:.x^(3)-3x^(2)-9x-5`
`=x^(2)(x+1)-4x^(2)-9x-5`
`=x^(2)(x+1)-4x(x+1)-5x-5`
`=x^(2)(x+1)-4x(x+1)-5(x+1)`
`=(x+1)(x^(2)-4x-5)`
`=(x+1)[x^(2)-5x+x-5]`
`=(x+1)[(x(x-5)+1(x-5)]`
`=(x+1)(x-5)(x+1)`
`=(x+1)^(2)(x-5)`
(iii) माना `p(x)=x^(3)+13x^(2)+32x+20`
यहाँ अचर पद `=20`
इसके गुणनखण्ड `=pm1,pm2,pm4,pm5,pm10,pm20`
`:." "p(-1)=(-1)^(3)+13(-1)^(2)+32(-1)+20`
`=-1+13-32+20=0`
`rArrx+1,p(x)` का एक गुणनखण्ड है |
`:.x^(3)+13x^(2)+32x+20`
`=x^(2)(x+1)+12x^(2)+32x+20`
`=x^(2)(x+1)+12x(x+1)+20x+20`
`=x^(2)(x+1)+12x(x+1)+20(x+1)`
`=(x+1)(x^(2)+12x+20)`
`=(x+1)[x^(2)+2x+10x+20]`
`=(x+1)[x(x+2)+10(x+2)]`
`=(x+1)(x+2)(x+10)`
(iv) माना `p(y)=2y^(3)+y^(2)-2y-1`
यहाँ अचर पद `=1`
इसके गुणनखण्ड `=pm1`
`:." "p(1)=2+1-2-1=0`
`rArry-1,p(y)` का एक गुणनखण्ड है |
`:." "2y^(3)+y^(2)-2y-1`
`=2y^(2)(y-1)+3y^(2)-2y-1`
`=2y^(2)(y-1)+3y(y-1)+y-1`
`=(y-1)(2y^(2)+3y+1)`
`(y-1)[2y^(2)+2y+y+1]`
`=(y-1)[2y(y+1)+1(y+1)]`
`=(y-1)(y+1)(2y+1)`