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The remainder theorem

  1. Feb 1, 2007 #1
    Find a polynomial of degree three that when divided by x - 2 has a remainder of 3. You will really have to think on this one. Hint: Work backwards!

    ok heres the thing i've tried i've looked at other problems but I can barely work problems forward, backwards...well your talking to me here my math skills are beyond horrible! my very sad attempt at this goes as follows

    x^3....???


    could someone just help at like walking me through it or tell me the first step i'll take anything.
     
  2. jcsd
  3. Feb 1, 2007 #2
    I'll see if I can get you to an understanding of the remainder theorem...

    Let's take the function f(x) = x^2 + 5x - 6
    substitute 1 for x. What do you get? 0
    Divide the function by (x-1). What's the remainder? 0
    f(x) = x^2 +5x -6 can be factored as (x-1)(x+6)
    That's why a value of 1 for x causes it to be zero.

    Now, let's review something else for a moment. Look at the graph of f(x) = x^2. Now, look at the graph of f(x) = x^2 + 1. Now, look at the graph of f(x) = x^2 +7. In the first graph, the vertex was at the origin. In the second graph, every point was "lifted up" one unit. In the 3rd graph, the original graph was shifted 7 units vertically. So, adding a number or subtracting a number at the end of a function serves to shift it up or down that many units.

    Now, let's look at f(x) = x^2 + 5x - 6 again. We noticed that when x = 1, the value of this function is 0. That's because (x-1) is a factor of the function. Thus, if you divided x^2 + 5x - 6 by (x-1), you'd get a remainder of 1.
    Now, for fun, let's add 17 to this function. f(x) = x^2 + 5x - 6 + 17, or
    f(x) = x^2 + 5x + 11. Plug in 1 for x, the same x value as last time. Hmmm... f(x) = 17. That shouldn't be a surprise; by adding 17 to the function, we took an original point on the function (1,0), and raised it 17 units to a new point: (1,17)

    Now, let's see what happens when we divide f(x)=x^2 + 5x +11 by (x-1). We get a remainder of 17! This also isn't a coincidence. Look at the quotient: (x + 6). By dividing, we have figured out that
    x^2 + 5x + 11 can be written as (x-1)(x+6) + a remainder of 17.
    Thus, f(x) = (x+1)(x+6) + 17
    This shows us that this new function (with +11 for the constant) is simply the old function raised 17 units.

    Now, start with a function of the order you desire; one that has a root of zero somewhere. Now, shift (translate) your function by adding a value to the constant. Check your answer.
     
  4. Feb 2, 2007 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    (x-2) times any second degree polynomial you want, plus 3!

    That's all you need to do.
     
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