Question

For what value of k the number -4 is a zero of the polynomial x²-x-(2k+2) and also find the other zero.

Answer 1

$$\begin{gathered} \mathrm{Let}p\left(x\right)=x^2-x-\left(2k+2\right)\mathrm{be}\mathrm{the}\mathrm{given}\mathrm{polynomial}. \\ \mathrm{Since},-4\mathrm{is}\mathrm{the}\mathrm{zero}\mathrm{of}\mathrm{the}\mathrm{polynomial}p\left(x\right),\mathrm{then} \\ p\left(-4\right)=0 \\ \Rightarrow {\left(-4\right)}^2-\left(-4\right)-\left(2k+2\right)=0 \\ \Rightarrow 16+4-2k-2=0 \\ \Rightarrow -2k+18=0 \\ \Rightarrow -2k=-18 \\ \Rightarrow \mathit{k}\mathbf{}\mathbf{=}\mathbf{}\mathbf{9} \\ \mathrm{putting}k=9\mathrm{in}p\left(x\right),\mathrm{we}\mathrm{get} \\ p\left(x\right)=x^2-x-20 \\ \mathrm{Now},\mathrm{we}\mathrm{know}\mathrm{that}\mathrm{zeroes}\mathrm{of}p\left(x\right)\mathrm{are}\mathrm{given}\mathrm{by}\mathrm{solving}\mathrm{the}\mathrm{equation}p\left(x\right)=0 \\ \mathrm{so}, \\ p\left(x\right)=0 \\ \Rightarrow x^2-x-20=0 \\ \Rightarrow x^2-5x+4x-20=0 \\ \Rightarrow x\left(x-5\right)+4\left(x-5\right)=0 \\ \Rightarrow \left(x-5\right)\left(x+4\right)=0 \\ \Rightarrow x-5=0orx+4=0 \\ \Rightarrow x=5orx=-4 \\ \mathrm{So},5\mathrm{is}\mathrm{the}\mathrm{other}\mathrm{zero}\mathrm{of}\mathrm{the}\mathrm{given}\mathrm{polynomial}. \end{gathered}$$