Tìm x ∈ Z để các phân số sau ∈ Z
a) $\frac{x-7}{x+4}$
b) $\frac{6x+4}{3x+1}$
c) $\frac{x+5}{x+3}$
d) $\frac{2x+7}{x-3}$
e) $\frac{6x-3}{3x+1}$
Tìm x ∈ Z để các phân số sau ∈ Z
a) $\frac{x-7}{x+4}$
b) $\frac{6x+4}{3x+1}$
c) $\frac{x+5}{x+3}$
d) $\frac{2x+7}{x-3}$
e) $\frac{6x-3}{3x+1}$
$\text{Đáp án + Giải thích các bước giải:}$
`a//(x-7)/(x+4)∈Z`
`=>x-7\vdots x+4`
`=>(x+4)-11\vdots x+4`
`=>11\vdots x+4`
`=>x+4∈Ư(11)={±1;±11}`
`=>x∈{-3;7;-5;-15}`
`b//(6x+4)/(3x+1)∈Z`
`=>6x+4\vdots 3x+1`
`=>2(3x+1)+2\vdots 3x+1`
`=>2\vdots 3x+1`
`=>3x+1∈Ư(2)={±1;±2}`
`=>3x∈{0;1;-2;-3}`
`=>x∈{0;-1}` $\text{. Do x ∈ Z}$
`c//(x+5)/(x+3)∈Z`
`=>x+5\vdots x+3`
`=>(x+3)+2\vdots x+3`
`=>2\vdots x+3`
`=>x+3∈Ư(2)={±1;±2}`
`=>x∈{-4;-5;-2;-1}`
`d//(2x+7)/(x-3)∈Z`
`=>2x+7\vdots x-3`
`=>2(x-3)+13\vdots x-3`
`=>13\vdots x-3`
`=>x-3∈Ư(13)={±1;±13}`
`=>x∈{4;16;2;-10}`
`e//(6x-3)/(3x+1)∈Z`
`=>6x-3\vdots 3x+1`
`=>2(3x+1)-5\vdots 3x+1`
`=>5\vdots 3x+1`
`=>3x+1∈Ư(5)={±1;±5}`
`=>3x∈{0;4;-2;-6}`
`=>x∈{0;-2}` $\text{. Do x ∈ Z }$
Đáp án:
`a)x\in \{-15;-5;-3;7\}`
`b)x\in\{-1;0\}`
`c)x\in \{-5;-4;-2;-1\}`
`d)x\in \{-10;2;4;16\}`
`e)x\in \{-2;0\}`
Giải thích các bước giải:
`a)` Để `\frac{x-7}{x+4}\in Z`
`=>x-7\vdots x+4`
`=>(x+4)-11\vdots x+4`
Do `x+4\vdots x+4`
`=>11\vdots x+4`
`=>x+4\in Ư(11)=\{-11;-1;1;11\}`
`=>x\in \{-15;-5;-3;7\}`
`b)` Để `\frac{6x+4}{3x+1}\in Z`
`=>6x+4\vdots 3x+1`
`=>6x+2+2\vdots 3x+1`
`=>2(3x+1)+2\vdots 3x+1`
Do `2(3x+1)\vdots 3x+1`
`=>2\vdots 3x+1`
`=>3x+1\in Ư(2)=\{-2;-1;1;2\}`
`=>3x\in \{-3;-2;0;1\}`
`=>x\in \{-1;0\}`
`c)` Để `\frac{x+5}{x+3}\in Z`
`=>x+5\vdots x+3`
`=>(x+3)+2\vdots x+3`
Do `x+3\vdots x+3`
`=>2\vdots x+3`
`=>x+3\in Ư(2)=\{-2;-1;1;2\}`
`=>x\in \{-5;-4;-2;-1\}`
`d)` Để `\frac{2x+7}{x-3}\in Z`
`=>2x+7\vdots x-3`
`=>2x-6+13\vdots x-3`
`=>2(x-3)+13\vdots x-3`
Do `2(x-3)\vdots x-3`
`=>13\vdots x-3`
`=>x-3\in Ư(13)=\{-13;-1;1;13\}`
`=>x\in \{-10;2;4;16\}`
`e)` Để `\frac{6x-3}{3x+1}\in Z`
`=>6x-3\vdots 3x+1`
`=>6x+2-5\vdots 3x+1`
`=>2(3x+1)-5\vdots 3x+1`
Do `2(3x+1)\vdots 3x+1`
`=>5\vdots 3x+1`
`=>3x+1\in Ư(5)=\{-5;-1;1;5\}`
`=>3x\in \{-6;-2;0;4\}`
`=>x\in \{-2;0\}`