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Divide x4 + 7 by x - 3.. A] x³ - 3x² - 9x - 27 R 88. B] x³ + 3x² + 9x - 27 R -74. C] x³ + 3x² + 9x + 27 R 88

Roger Moore

in Mathematics

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Timothy Norman on November 5, 2018

divide a polynomial p(x) by (x-3). Add and subtract a multiple of (x-3) that has the same highest order term as p(x), then simplify to obtain a lower degree of the polynomial r(x), in addition to multiple of (x-3). The multiple of (x-3) which is x^4 as its leading term is x^3(x-3) = x^4 - 3x^3. So write: x^4 + 7 = x^4 + 7 + x^3(x - 3) - x^3(x - 3) = x^4 + 7 + x^3(x - 3) - x^4 + 3x^3 = x^3(x - 3) + 3x^3 + 7 That makes r(x) = 3x^3 + 7. Do the same to reduce r(x) by the sum/subtract 3x^2(x - 3) = 3x^3 - 9x^2: = x^3(x - 3) + 3x^3 + 7 + 3x^2(x - 3) - (3x^3 - 9x^2) = x^3(x - 3) + 3x^2(x - 3) + 9x^2 + 7 Again to reduce 9x^2 + 7: = x^3(x - 3) + 3x^2(x - 3) + 9x^2 + 7 + 9 x(x - 3) - (9x^2 - 27x) = x^3(x - 3) + 3x^2(x - 3) + 9x(x - 3) + 27x + 7 And finally write 27x + 7 27(x - 3) + 88; x^4 + 7 = x^3(x - 3) + 3x^2(x - 3) + 9x(x - 3) + 27(x - 3) + 88 Factor (x - 3) in all but the +88 term: x^4 + 7 = (x - 3)(x^3 + 3x^2 + 9x + 27) + 88, Which means that: (x^4 + 7) / (x - 3) = x^3 + 3x^2 + 9x + 27 with a remainder of 88


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