Dùng định nghĩa hai phân thức bằng nhau, hãy tìm đa thức A biết 5x 2 − 13x + 6 A = 5x − 3 2x + 5 ?
A . A = 2 x 2 + x + 10
B . A = 2 x 2 + x - 10
C . A = 2 x 2 - x - 10
D . A = x 2 + x - 10
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\(\frac{5x^2-13x+6}{A}=\frac{5x-3}{2x+5}\)
\(\Rightarrow\left(5x^2-13x+6\right)\left(2x+5\right)=A.\left(5x-3\right)\)
\(\Rightarrow\left[5x\left(x-2\right)-3\left(x-2\right)\right]\left(2x+5\right)=A.\left(5x-3\right)\)
\(\Rightarrow\left(x-2\right)\left(5x-3\right)\left(2x+5\right)=A.\left(5x-3\right)\)
\(\Rightarrow A=\left(x-2\right)\left(2x+5\right)=2x^2+x-10\)
\(\dfrac{A}{x-3}=\dfrac{y-x}{3-x}\)
\(\Rightarrow A=\dfrac{\left(x-3\right)\left(y-x\right)}{3-x}\)
\(\Rightarrow A=\dfrac{-\left(3-x\right)\left(y-x\right)}{3-x}\)
\(\Rightarrow A=x-y\)
_____
\(\dfrac{5x}{x+1}=\dfrac{Ax\left(x+1\right)}{\left(1-x\right)\left(1+x\right)}\)
\(\Rightarrow A=\dfrac{5x\left(x+1\right)\left(1-x\right)}{x\left(x+1\right)}\)
\(\Rightarrow A=5\left(1-x\right)\)
\(\Rightarrow A=5-5x\)
____
\(\dfrac{4x^2-5x+1}{A}=\dfrac{4x-1}{x+3}\)
\(\Rightarrow\dfrac{\left(4x-1\right)\left(x-1\right)}{A}=\dfrac{4x-1}{x+3}\)
\(\Rightarrow A=\dfrac{\left(4x-1\right)\left(x-1\right)\left(x+3\right)}{4x-1}\)
\(\Rightarrow A=\left(x-1\right)\left(x+3\right)\)
\(\Rightarrow A=x^2+2x-3\)
⇒ 4 x 2 - 7 x + 3 x 2 + 2 x + 1 = A x 2 - 1
⇒ 4 x 2 - 4 x - 3 x + 3 x + 1 2 = A x + 1 x - 1
⇒ 4 x x - 1 - 3 x - 1 . x + 1 2 = A . x + 1 x - 1
⇒ x - 1 4 x - 3 x + 1 2 = A x + 1 x - 1
⇒ A = 4 x - 3 x + 1 = 4 x 2 + 4 x - 3 x - 3 = 4 x 2 + x - 3
Vậy
d: \(\Leftrightarrow\dfrac{\left(x+2\right)^2}{\left(x+2\right)\left(x-2\right)}=\dfrac{\left(x+1\right)\left(x+2\right)}{A}\)
hay A=x-2
⇒ 4 x 2 - 3 x - 7 2 x + 3 = A 4 x - 7
⇒ 4 x 2 + 4 x - 7 x - 7 2 x + 3 = A 4 x - 7
⇒ [4x(x + 1) – 7(x + 1)](2x+ 3) = A(4x - 7)
⇒ (x + 1)(4x – 7)(2x + 3) = A(4x – 7)
⇒ A = (x + 1)(2x + 3) = 2 x 3 + 3 x + 2 x + 3 = 2 x 2 + 5 x + 3
Vậy
= ( 5 x - 3 ) ( x - 2 ) ( 2 x + 5 ) : ( 5 x - 3 )