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mcshweezy

What is the remainder when (3x4 + 2x3 – x2 + 2x – 14) ÷ (x + 2) ?

  • 2 years ago
  • 2 years ago

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  1. hoblos
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    10

    • 2 years ago
  2. JamesJ
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    @hoblos: if you answer it, you explain it!

    • 2 years ago
  3. mathmate
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    What you do is to evaluate the numerator, using x=-2 (from x+2=0), and what you get is the remainder, namely 10 as Hoblos posted.

    • 2 years ago
  4. cwrw238
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    this is called the Remainder THEorem

    • 2 years ago
  5. jhonyy9
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    (3x4+2x3-x2+2x-14) : (x+2) =3x3-4x2+7x-12 3x4+6x3 --------- 0 -4x3-x2 -4x3-8x2 ---------- 0 7x2+2x 7x2+14x ---------- 0 -12x-14 -12x-24 --------- 0 10

    • 2 years ago
  6. JamesJ
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    @cwrw: right. The idea is that given any polynomial p(x) and a linear term such as (x+2), then we write p(x) as p(x) = (x+2)q(x) + r --- (*) where q(x) is another polynomial. r is the remainder after p(x) is divided by (x+2). ===== If x=-2 is a root of the polynomial p(x), then p(-2) = 0. Hence using equation (*), 0 = p(-2) = (-2+2)q(-2) + r = 0 + r i.e., r = 0. If x=-2 is not a root of p(x), then r will not be zero. It can be calculated by evaluating p(-2), because p(-2) = (-2+2)q(-2) + r = 0 + r i.e., r = p(-2) ===== So this is why a short-cut way to finding the remainder of p(x) = 3x^4 + 2x^3 – x^2 + 2x – 14 when divided by (x+2) is just to evaluate p(x) for x = -2: r = p(-2) = 3(-2)^4 + 2(-2)^3 - (-2)^2 + 2(-2) - 14 = 48 - 16 - 4 - 4 - 14 = 10

    • 2 years ago
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