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Question 5 The expression x^(3)+2x^(2)+px-3 has the same remainder when it is divided by x+1 and by x -2 . Find the value of p. [3]

Question 55\newlineThe expression x3+2x2+px−3 x^{3}+2 x^{2}+p x-3 has the same remainder when it is divided by x+1 x+1 and by x x −2-2 . Find the value of p p .\newline[33]

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Q. Question 55\newlineThe expression x3+2x2+px−3 x^{3}+2 x^{2}+p x-3 has the same remainder when it is divided by x+1 x+1 and by x x −2-2 . Find the value of p p .\newline[33]
  1. Substitute and Simplify: To find the remainder when the polynomial is divided by x+1x + 1, substitute −1-1 for xx in the polynomial.\newlineSo, (−1)3+2(−1)2+p(−1)−3=−1+2−p−3(-1)^3 + 2(-1)^2 + p(-1) - 3 = -1 + 2 - p - 3.
  2. Find Remainder for x+1x + 1: Simplify the expression to find the remainder for x+1x + 1. The remainder is −1+2−p−3=−2−p-1 + 2 - p - 3 = -2 - p.
  3. Substitute and Simplify: To find the remainder when the polynomial is divided by x−2x - 2, substitute 22 for xx in the polynomial.\newlineSo, 23+2(2)2+p(2)−3=8+8+2p−32^3 + 2(2)^2 + p(2) - 3 = 8 + 8 + 2p - 3.
  4. Find Remainder for x−2x - 2: Simplify the expression to find the remainder for x−2x - 2. The remainder is 8+8+2p−3=13+2p8 + 8 + 2p - 3 = 13 + 2p.
  5. Set Equations Equal: Since the remainders must be equal, set the two expressions equal to each other.\newline−2−p=13+2p-2 - p = 13 + 2p.
  6. Solve for pp: Solve for pp by adding pp to both sides and subtracting 1313 from both sides.\newline−2−p+p+13=13+2p+p−13-2 - p + p + 13 = 13 + 2p + p - 13.

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