Solve (x-1)quad(x^2+mx+n)=x^3-6x^2+11x-6

$$( x - 1 ) \quad ( x ^ { 2 } + m x + n ) = x ^ { 3 } - 6 x ^ { 2 } + 11 x - 6$$

Solve for m (complex solution)

$$\left\{\begin{matrix}m=-\frac{5x+n-6}{x}\text{, }&x\neq 0\\m\in \mathrm{C}\text{, }&x=1\text{ or }\left(x=0\text{ and }n=6\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-1 by x^{2}+mx+n.

$$x^{3}+mx^{2}+xn-x^{2}-mx-n=x^{3}-6x^{2}+11x-6$$

Subtract x^{3} from both sides.

$$mx^{2}+xn-x^{2}-mx-n=x^{3}-6x^{2}+11x-6-x^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$mx^{2}+xn-x^{2}-mx-n=-6x^{2}+11x-6$$

Subtract xn from both sides.

$$mx^{2}-x^{2}-mx-n=-6x^{2}+11x-6-xn$$

Add x^{2} to both sides.

$$mx^{2}-mx-n=-6x^{2}+11x-6-xn+x^{2}$$

Combine -6x^{2} and x^{2} to get -5x^{2}.

$$mx^{2}-mx-n=-5x^{2}+11x-6-xn$$

Add n to both sides.

$$mx^{2}-mx=-5x^{2}+11x-6-xn+n$$

Combine all terms containing m.

$$\left(x^{2}-x\right)m=-5x^{2}+11x-6-xn+n$$

The equation is in standard form.

$$\left(x^{2}-x\right)m=-5x^{2}-nx+11x+n-6$$

Divide both sides by x^{2}-x.

$$\frac{\left(x^{2}-x\right)m}{x^{2}-x}=\frac{\left(1-x\right)\left(5x+n-6\right)}{x^{2}-x}$$

Dividing by x^{2}-x undoes the multiplication by x^{2}-x.

$$m=\frac{\left(1-x\right)\left(5x+n-6\right)}{x^{2}-x}$$

Divide \left(-6+5x+n\right)\left(1-x\right) by x^{2}-x.

$$m=-\frac{5x+n-6}{x}$$

Solve for n (complex solution)

$$\left\{\begin{matrix}\\n=6-5x-mx\text{, }&\text{unconditionally}\\n\in \mathrm{C}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-1 by x^{2}+mx+n.

$$x^{3}+mx^{2}+xn-x^{2}-mx-n=x^{3}-6x^{2}+11x-6$$

Subtract x^{3} from both sides.

$$mx^{2}+xn-x^{2}-mx-n=x^{3}-6x^{2}+11x-6-x^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$mx^{2}+xn-x^{2}-mx-n=-6x^{2}+11x-6$$

Subtract mx^{2} from both sides.

$$xn-x^{2}-mx-n=-6x^{2}+11x-6-mx^{2}$$

Add x^{2} to both sides.

$$xn-mx-n=-6x^{2}+11x-6-mx^{2}+x^{2}$$

Add mx to both sides.

$$xn-n=-6x^{2}+11x-6-mx^{2}+x^{2}+mx$$

Combine -6x^{2} and x^{2} to get -5x^{2}.

$$xn-n=-5x^{2}+11x-6-mx^{2}+mx$$

Combine all terms containing n.

$$\left(x-1\right)n=-5x^{2}+11x-6-mx^{2}+mx$$

The equation is in standard form.

$$\left(x-1\right)n=-mx^{2}-5x^{2}+mx+11x-6$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)n}{x-1}=\frac{\left(x-1\right)\left(6-5x-mx\right)}{x-1}$$

Dividing by x-1 undoes the multiplication by x-1.

$$n=\frac{\left(x-1\right)\left(6-5x-mx\right)}{x-1}$$

Divide \left(-1+x\right)\left(6-5x-xm\right) by x-1.

$$n=6-5x-mx$$

Solve for m

$$\left\{\begin{matrix}m=-\frac{5x+n-6}{x}\text{, }&x\neq 0\\m\in \mathrm{R}\text{, }&x=1\text{ or }\left(x=0\text{ and }n=6\right)\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-1 by x^{2}+mx+n.

$$x^{3}+mx^{2}+xn-x^{2}-mx-n=x^{3}-6x^{2}+11x-6$$

Subtract x^{3} from both sides.

$$mx^{2}+xn-x^{2}-mx-n=x^{3}-6x^{2}+11x-6-x^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$mx^{2}+xn-x^{2}-mx-n=-6x^{2}+11x-6$$

Subtract xn from both sides.

$$mx^{2}-x^{2}-mx-n=-6x^{2}+11x-6-xn$$

Add x^{2} to both sides.

$$mx^{2}-mx-n=-6x^{2}+11x-6-xn+x^{2}$$

Combine -6x^{2} and x^{2} to get -5x^{2}.

$$mx^{2}-mx-n=-5x^{2}+11x-6-xn$$

Add n to both sides.

$$mx^{2}-mx=-5x^{2}+11x-6-xn+n$$

Combine all terms containing m.

$$\left(x^{2}-x\right)m=-5x^{2}+11x-6-xn+n$$

The equation is in standard form.

$$\left(x^{2}-x\right)m=-5x^{2}-nx+11x+n-6$$

Divide both sides by x^{2}-x.

$$\frac{\left(x^{2}-x\right)m}{x^{2}-x}=\frac{\left(1-x\right)\left(5x+n-6\right)}{x^{2}-x}$$

Dividing by x^{2}-x undoes the multiplication by x^{2}-x.

$$m=\frac{\left(1-x\right)\left(5x+n-6\right)}{x^{2}-x}$$

Divide \left(-6+5x+n\right)\left(1-x\right) by x^{2}-x.

$$m=-\frac{5x+n-6}{x}$$

Solve for n

$$\left\{\begin{matrix}\\n=6-5x-mx\text{, }&\text{unconditionally}\\n\in \mathrm{R}\text{, }&x=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply x-1 by x^{2}+mx+n.

$$x^{3}+mx^{2}+xn-x^{2}-mx-n=x^{3}-6x^{2}+11x-6$$

Subtract x^{3} from both sides.

$$mx^{2}+xn-x^{2}-mx-n=x^{3}-6x^{2}+11x-6-x^{3}$$

Combine x^{3} and -x^{3} to get 0.

$$mx^{2}+xn-x^{2}-mx-n=-6x^{2}+11x-6$$

Subtract mx^{2} from both sides.

$$xn-x^{2}-mx-n=-6x^{2}+11x-6-mx^{2}$$

Add x^{2} to both sides.

$$xn-mx-n=-6x^{2}+11x-6-mx^{2}+x^{2}$$

Add mx to both sides.

$$xn-n=-6x^{2}+11x-6-mx^{2}+x^{2}+mx$$

Combine -6x^{2} and x^{2} to get -5x^{2}.

$$xn-n=-5x^{2}+11x-6-mx^{2}+mx$$

Combine all terms containing n.

$$\left(x-1\right)n=-5x^{2}+11x-6-mx^{2}+mx$$

The equation is in standard form.

$$\left(x-1\right)n=-mx^{2}-5x^{2}+mx+11x-6$$

Divide both sides by x-1.

$$\frac{\left(x-1\right)n}{x-1}=\frac{\left(x-1\right)\left(6-5x-mx\right)}{x-1}$$

Dividing by x-1 undoes the multiplication by x-1.

$$n=\frac{\left(x-1\right)\left(6-5x-mx\right)}{x-1}$$

Divide \left(-1+x\right)\left(6-5x-xm\right) by x-1.

$$n=6-5x-mx$$