Question

If the zeros of the polynomial f(x) = x³ − 3x² + x + 1 are (a − b), a and (a + b), Find a and b.

Answer 1

By using the relationship between the zeroes of the cubic ploynomial.
We have, Sum of zeroes = $\frac{-\left(\mathrm{coefficient}\mathrm{of}\mathit{}{x}^2\right)}{\mathrm{coefficent}\mathrm{of}{x}^3}$
$$\begin{gathered} \therefore a-b+a+a+b=\frac{-\left(-3\right)}{1} \\ \Rightarrow 3a=3 \\ \Rightarrow a=1 \end{gathered}$$

Now, Product of zeros = $\frac{-\left(\mathrm{constant}\mathrm{term}\right)}{\mathrm{coefficent}\mathrm{of}{x}^3}$
$$\begin{gathered} \therefore \left(a-b\right)\left(a\right)\left(a+b\right)=\frac{-1}{1} \\ \Rightarrow \left(1-b\right)\left(1\right)\left(1+b\right)=-1\left[\because a=1\right] \\ \Rightarrow 1-b^2=-1 \\ \Rightarrow b^2=2 \\ \Rightarrow b=\pm \sqrt{2} \end{gathered}$$