given distinct real numbersx1 x2 xn definepx xx1xx2xxnand for each1jn definepjx pxxxj so that al

Name: given distinct real numbersx1 x2 xn definepx xx1xx2xxnand for each1jn definepjx pxxxj so that all the numbersx1 x2 xnexceptxjare roots ofpjx problem2 given distinct real numbers 12ndefine p 05964
Uploaded: 2023-07-25T13:16:07-08:00
Duration: 21 s
Channel: Numerade
Description: given distinct real numbersx1 x2 xn definepx xx1xx2xxnand for each1jn definepjx pxxxj so that all the numbersx1 x2 xnexceptxjare roots ofpjx problem2 given distinct real numbers 12ndefine p 05964

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Okay, so here we have our polynomial p of x equal to x to the 2n plus 1 plus ax plus b And we kn

Question

Given distinct real numbers x1, x2, ..., xn, define P(x) = (x - x1)(x - x2)...(x - xn); and for each 1 â‰¤ j â‰¤ n, define Pj(x) = P(x)/(x - xj), so that all the numbers x1, x2, ..., xn except xj are roots of Pj(x). Problem 2. Given distinct real numbers 1, 2, ..., n, define P(x) = (x - 1)(x - 2)...(x - n); and for each 1 â‰¤ j â‰¤ n, define Pj(x) = P(x)/(x - j), so that all the numbers 1, 2, ..., n except j are roots of Pj(x). (a) If Q(x) is any polynomial of degree less than n, prove that P(x)Q(x)/Pn(n) = Î£(j=1 to n) Q(xj)/(x - xj). (b) Use the formula from part (a) to find a cubic polynomial Q such that Q(1) = 117, Q(2) = 239, Q(3) = 369, and Q(4) = 513. Write your answer in the standard form ar^3 + bx^2 + cx + d.

          Given distinct real numbers x1, x2, ..., xn, define P(x) = (x - x1)(x - x2)...(x - xn); and for each 1 â‰¤ j â‰¤ n, define Pj(x) = P(x)/(x - xj), so that all the numbers x1, x2, ..., xn except xj are roots of Pj(x).

Problem 2. Given distinct real numbers 1, 2, ..., n, define P(x) = (x - 1)(x - 2)...(x - n); and for each 1 â‰¤ j â‰¤ n, define Pj(x) = P(x)/(x - j), so that all the numbers 1, 2, ..., n except j are roots of Pj(x).

(a) If Q(x) is any polynomial of degree less than n, prove that P(x)Q(x)/Pn(n) = Î£(j=1 to n) Q(xj)/(x - xj).

(b) Use the formula from part (a) to find a cubic polynomial Q such that Q(1) = 117, Q(2) = 239, Q(3) = 369, and Q(4) = 513. Write your answer in the standard form ar^3 + bx^2 + cx + d.

given distinct real numbersx1 x2 xn definepx xx1xx2xxnand for each1jn definepjx pxxxj so that all the numbersx1 x2 xnexceptxjare roots ofpjx problem2 given distinct real numbers 12ndefine p 05964

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