If the zeroes of the polynomial $x^{3} - 3 x^{2} + x + 1$ are m – n, m and m + n, find m and n.

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Solution

We know that for the cubic polynomials, sum of the zeroes = $\frac{- b}{a}$

m – n + m + m + n = 3 $\Rightarrow$ 3m = 3 $\Rightarrow$ m = 1

Sum of the product of the zeroes (taken 2 at a time) = $\frac{c}{a}$

$(m - n) \times m + m \times (m + n) + (m + n) \times (m - n) = 1 \Rightarrow 3 m^{2} - n^{2} = 1 \Rightarrow n^{2} = 2 n = \pm \sqrt{2}$

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If the zeroes of the polynomial x3−3x2+x+1 are m – n, m and m + n, find m and n.

We know that for the cubic polynomials, sum of the zeroes = −ba m – n + m + m + n = 3 ⇒ 3m = 3 ⇒ m = 1 Sum of the product of the zeroes (taken 2 at a time) =ca (m−n)×m+m×(m+n)+(m+n)×(m−n)=1⇒3m2−n2=1⇒n2=2n=±√2

If the zeroes of the polynomial $x^{3} - 3 x^{2} + x + 1$ are m – n, m and m + n, find m and n.