Question

Find the value of the polynomial y³ $-$ 5y $-$ 2y² + 3 when.

(i) y = 1
(ii) y = 2
(iii) y = $-$ 2
(iv) y = 4
(v) y = $-$ b

Answer 1

Let p(y) = $y^3-5y-2y^2+3$

$$\begin{gathered} \left(\mathrm{i}\right)\mathrm{To}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}p\left(y\right)\mathrm{when}y=1,\mathrm{put}y=1\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{polynomial}. \\ p\left(y\right)=y^3-5y-2y^2+3 \\ \Rightarrow p\left(1\right)=1^3-5\times 1-2\times 1^2+3 \\ =1-5-2+3 \\ =-3 \\ \Rightarrow p\left(1\right)=-3 \\ \therefore \mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{polynomial}\mathrm{is}-3\mathrm{when}y=1 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\mathrm{To}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}p\left(y\right)\mathrm{when}y=2,\mathrm{put}y=2\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{polynomial}. \\ p\left(y\right)=y^3-5y-2y^2+3 \\ \Rightarrow p\left(2\right)=2^3-5\times 2-2\times 2^2+3 \\ =8-10-8+3 \\ =-7 \\ \Rightarrow p\left(2\right)=-7 \\ \therefore \mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{polynomial}\mathrm{is}-7\mathrm{when}y=2 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\mathrm{To}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}p\left(y\right)\mathrm{when}y=-2,\mathrm{put}y=-2\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{polynomial}. \\ p\left(y\right)=y^3-5y-2y^2+3 \\ \Rightarrow p\left(-2\right)={\left(-2\right)}^3-5\times \left(-2\right)-2\times {\left(-2\right)}^2+3 \\ =-8+10-8+3 \\ =-3 \\ \Rightarrow p\left(-2\right)=-3 \\ \therefore \mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{polynomial}\mathrm{is}-3\mathrm{when}y=-2 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)\mathrm{To}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}p\left(y\right)\mathrm{when}y=4,\mathrm{put}y=4\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{polynomial}. \\ p\left(y\right)=y^3-5y-2y^2+3 \\ \Rightarrow p\left(4\right)=4^3-5\times 4-2\times 4^2+3 \\ =64-20-32+3 \\ =15 \\ \Rightarrow p\left(4\right)=15 \\ \therefore \mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{polynomial}\mathrm{is}15\mathrm{when}y=4 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right)\mathrm{To}\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}p\left(y\right)\mathrm{when}y=-b,\mathrm{put}y=-b\mathrm{in}\mathrm{the}\mathrm{given}\mathrm{polynomial}. \\ p\left(y\right)=y^3-5y-2y^2+3 \\ \Rightarrow p\left(-b\right)={\left(-b\right)}^3-5\times \left(-b\right)-2\times {\left(-b\right)}^2+3 \\ =-b^3+5b-2b^2+3 \\ =-b^3-2b^2+5b+3 \\ \Rightarrow p\left(-b\right)=-b^3-2b^2+5b+3 \\ \therefore \mathrm{The}\mathrm{value}\mathrm{of}\mathrm{the}\mathrm{polynomial}\mathrm{is}-b^3-2b^2+5b+3\mathrm{when}y=-b \end{gathered}$$