Polynomials, EXT1 EQ-Bank 16 The polynomial \(R(x)=x^3+p x^2+q x+6\) has a double zero at \(x=-1\) and a zero at \(x=s\). Find the values of \(p, q\) and \(s\). (3 marks) --- 10 WORK AREA LINES (style=lined) --- Show Answers Only \(s=-6, \ p=8, \ q=13\) Show Worked Solution \(R(x)=x^3+p x^2+q x+6\) \(R(x)\ \text{is monic with a zero at} \ s \ \text{and double zero at}\ -1:\) \(R(x)\) \(=(x+1)^2(x-s)\) \(=\left(x^2+2 x+1\right)(x-s)\) \(=x^3+2 x^2+x-s x^2-2 s x-s\) \(=x^3+(2-s) x^2+(1-2 s) x-s\) \(\text{Equating coefficients:}\) \(-s=6 \ \Rightarrow \ s=-6\) \(p=2-(-6)=8\) \(q=1-2(-6)=13\)