Question

Tìm x nguyên để P là số nguyên:

a) P=\(\dfrac{2x+5}{x+3}\)

b) P=\(\dfrac{3x+4}{x+1}\)

c) P=\(\dfrac{4x-1}{2x+3}\)

Answer 1

a) \(P=\dfrac{2x+5}{x+3}\inℤ\left(x\inℤ;x\ne-3\right)\)

\(\Rightarrow2x+5⋮x+3\)

\(\Rightarrow2x+5-2\left(x+3\right)⋮x+3\)

\(\Rightarrow2x+5-2x-6⋮x+3\)

\(\Rightarrow-1⋮x+3\)

\(\Rightarrow x+3\in\left\{-1;1\right\}\)

\(\Rightarrow x\in\left\{-4;-2\right\}\)

b) \(P=\dfrac{3x+4}{x+1}\inℤ\left(x\inℤ;x\ne-1\right)\)

\(\Rightarrow3x+4⋮x+1\)

\(\Rightarrow3x+4-3\left(x+1\right)⋮x+1\)

\(\Rightarrow3x+4-3x-3⋮x+1\)

\(\Rightarrow1⋮x+1\)

\(\Rightarrow x+1\in\left\{-1;1\right\}\)

\(\Rightarrow x\in\left\{-2;0\right\}\)

c) \(P=\dfrac{4x-1}{2x+3}\inℤ\left(x\inℤ;x\ne-\dfrac{3}{2}\right)\)

\(\Rightarrow4x-1⋮2x+3\)

\(\Rightarrow4x-1-2\left(2x+3\right)⋮2x+3\)

\(\Rightarrow4x-1-4x-6⋮2x+3\)

\(\Rightarrow-7⋮2x+3\)

\(\Rightarrow2x+3\in\left\{-1;1;-7;7\right\}\)

\(\Rightarrow x\in\left\{-2;-1;-5;2\right\}\)

Answer 2

a) P=\(\dfrac{2x+5}{x+3}=\dfrac{2\left(x+3\right)-2}{x+3}=\dfrac{2\left(x+3\right)}{x+3}-\dfrac{2}{x+3}=2-\dfrac{2}{x+3}\)

để \(P\inℤ\) thì \(\dfrac{2}{x+3}\inℤ\) hay 2 ⋮ (x-3) ⇒x+3 ϵ Ư2= (2,-2,1,-1)

ta có bảng sau:

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

Vậy x \(\in-1,-2,-5,-4\)