For a homogeneous polynomial with real coefficients:$f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz$, suppose we know $f$ factors into products of linear forms$f=(p_1x+p_2y+p_3z)(q_1x+q_2y+q_3z)$, are there anyway to determinate whether the coefficients of the linear forms are real or complex?

I know we could always plug in (1,1,0) and (1,0,1) into (x,y,z) to reduce the problem into binary forms. Are there any neater test than that?

For a homogeneous polynomial with real coefficients:$f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz$, suppose we know $f$ factors into products of linear forms$f=(p_1x+p_2y+p_3z)(q_1x+q_2y+q_3z)$, are there anyway to determinate whether the coefficients of the linear forms are real or complex?

For a homogeneous polynomial with real coefficients:$f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz$, suppose we know $f$ factors into products of linear forms$f=(p_1x+p_2y+p_3z)(q_1x+q_2y+q_3z)$, are there anyway to determinate whether the coefficients of the linear forms are real or complex?

For a homogeneous polynomial with real coefficients:$f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz$, suppose we know $f$ factors into products of linear forms$f=(p_1x+p_2y+p_3z)(q_1x+q_2y+q_3z)$, are there anyway to determinate whether the coefficients of the linear forms are real or complex?

I know we could always plug in (1,1,0) and (1,0,1) into (x,y,z) to reduce the problem into binary forms. Are there any neater test than that?