Question 25: Find the coefficients a, b, c knowing \((\operatorname{ax}+b) \cdot\left(x^{2}-c x+2\right)=x^{3}+x ^{2}-2 \text { true for all x. }\)
We have
\(\begin{aligned} &(a x+b) \cdot\left(x^{2}-c x+2\right)=x^{3}+x^{2}-2\\ &\ Leftrightarrow ax^{3}+bx^{2}-acx^{2}-bc x+2 b+2 ax=x^{3}+x^{2}-2\\ &\Leftrightarrow \mathrm{ax }^{3}+(three c) x^{2}+(2 ab c) x+2 b=x^{3}+x^{2}-2
\\ &\text { true for all } \mathrm{x}\\ &\Leftrightarrow\left\{\begin{array} { l } { a = 1 } \\ { 2 b = – 2 } \\ { b – ac = 1 } \ \ { 2 a – bc = 0 } \end{array} \Leftrightarrow \left\{\begin{array} { l } { a = 1 } \\ { b = – 1 } \\ { – 1 – 1 . c = 1 } \\ { 2 – ( – 1 ) \cdot c = 0 } \end{array} \Leftrightarrow \left\{\begin{array}{l} a=1 \\ b=-1 \\ c =-2 \end{array}\right.\right.\right. \end{aligned}\)
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