Answer :

Given, p(1) = 0, p(√2) = 0 and p(– √2) = 0


Need to find the third degree polynomial p(x)


p(1) = 0 is given which satisfy p(x)


if x = 1 is substituted in p(x) then it satisfies the equation


x – 1 is one factor of p(x)


p(√2) = 0 is given


if x = √2 is substituted in p(x) then it satisfies the equation


x – √2 is another factor


p(– √2) = 0 is given


if x = – √2 is substituted in p(x) then it satisfies the equation


x + √2 is third factor of the p(x)


Since, (x – 1)(x – √2)(x + √2) are the factors of the third degree polynomials


p(x) = (x – 1)(x – √2)(x + √2)


p(x) = (x2 – x – √2x + √2)(x + √2)


p(x) = (x3 + √2x2 – x2 – √2x – √2x2 – 2x + √2x + 2)


p(x) = (x3 – x2 – 2x + 2)


Hence, x3 – x2 – 2x + 2 is the third degree polynomial which satisfies


p(1) = 0, p(√2) = 0 and p(– √2) = 0


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