The cubic polynomial P(x) =x3 +rx2 +sx +t
, where r, s and t are real numbers, has three real zeros, 1, αand –α.
(i) Find the value of r.
(ii) Find the value of s +t.
P(x) =x3 +rx2 +sx +t.

The cubic polynomial P(x) =x3 +rx2 +sx +t
, where r, s and t are real numbers, has three real zeros, 1, αand –α.
(i) Find the value of r.
(ii) Find the value of s +t.
P(x) =x3 +rx2 +sx +t.

Write \(\displaystyle P(x) = (x-1)(x-\alpha)(x-(-\alpha))\), expand, and then match the coefficients.

Note that \(\displaystyle (x-\alpha)(x+\alpha)=x^2-\alpha^2\) (in other words, you can skip the step of writing out that expansion explicitly by noticing there is a difference of squares).

Write \(\displaystyle P(x) = (x-1)(x-\alpha)(x-(-\alpha))\), expand, and then match the coefficients.

Note that \(\displaystyle (x-\alpha)(x+\alpha)=x^2-\alpha^2\) (in other words, you can skip the step of writing out that expansion explicitly by noticing there is a difference of squares).

Almost. r = -1, not 1. I hope you see how I matched the coefficients, because that's the main point of the problem, and you might get more problems like it.

Almost. r = -1, not 1. I hope you see how I matched the coefficients, because that's the main point of the problem, and you might get more problems like it.