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Math Help - Help with factoring cubic polynomials and find the zeros of the function

  1. #1
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    Help with factoring cubic polynomials and find the zeros of the function

    Yikes! I spent hours on this problem w/o any result.

    f(x) = x^3 - x^2 - 2x + 2 = 0

    My first step was to plug in some numbers and see if any of them would result in a zero.

    f(x) = 0

    I plugged in 1 and it resulted in a zero. So I figured that (x+1) is one of my factors. I proceeded to do synthetic division and that's when things got crazy.

    How do I figure out the remaining factors? Please help!

    Please see the attached. Thanks!
    Attached Thumbnails Attached Thumbnails Help with factoring cubic polynomials and find the zeros of the function-scan0001.jpg  
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  2. #2
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    Quote Originally Posted by lilrhino View Post
    Yikes! I spent hours on this problem w/o any result.

    f(x) = x^3 - x^2 - 2x + 2 = 0

    My first step was to plug in some numbers and see if any of them would result in a zero.

    f(x) = 0

    I plugged in 1 and it resulted in a zero. So I figured that (x+1) is one of my factors. I proceeded to do synthetic division and that's when things got crazy.
    (x-1) is a factor, so try:

    x^3 - x^2 - 2x + 2 = (x-1)(x^2 + ax -2)

    The linear term on the right is then -a-2x, which should be equal to that
    on the left, so a=0.

    Now check by multiplying out the following:

    (x-1)(x^2-2)

    Now (x^2-2)=(x-sqrt(2))(x+sqrt(2), so:

    x^3 - x^2 - 2x + 2 = (x-1)(x-sqrt(2))(x+sqrt(2)).

    RonL
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    (x-1) is a factor, so try:

    x^3 - x^2 - 2x + 2 = (x-1)(x^2 + ax -2)

    The linear term on the right is then -a-2x, which should be equal to that
    on the left, so a=0.

    Now check by multiplying out the following:

    (x-1)(x^2-2)

    Now (x^2-2)=(x-sqrt(2))(x+sqrt(2), so:

    x^3 - x^2 - 2x + 2 = (x-1)(x-sqrt(2))(x+sqrt(2)).

    RonL
    Thanks RonL, but I'm still trying to comprehend the logic around "the linear term -a-2x should equal the term on the left, so a=0".

    Can you provide a little more detail as to why this is used to solve and why the terms should equal one another on both sides?

    It appears as if you found terms to eliminate the others so you could end with:

    (x-1)(x^2-2)

    I know you're a math whiz, what how would a novice like myself know to do that? What is the trigger to know that I should perform these particular steps? Sorry for all the questions, but I really want to understand this. Thanks again!
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  4. #4
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    Quote Originally Posted by lilrhino View Post
    Thanks RonL, but I'm still trying to comprehend the logic around "the linear term -a-2x should equal the term on the left, so a=0".

    Can you provide a little more detail as to why this is used to solve and why the terms should equal one another on both sides?

    It appears as if you found terms to eliminate the others so you could end with:

    (x-1)(x^2-2)

    I know you're a math whiz, what how would a novice like myself know to do that? What is the trigger to know that I should perform these particular steps? Sorry for all the questions, but I really want to understand this. Thanks again!
    Just multiply out the right hand side of:

    x^3 - x^2 - 2x + 2 = (x-1)(x^2 + ax -2)

    x^3 - x^2 - 2x + 2 = x^3 +x^2(a-1) +x(-2-a) +2

    Now the coefficients of like powers of x on both sides of this equation
    must be equal, so in particular the coeficients of the linear terms are equal
    so:

    -2 = -2-a

    and so a=0.

    RonL
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  5. #5
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    Quote Originally Posted by CaptainBlack View Post
    Just multiply out the right hand side of:

    x^3 - x^2 - 2x + 2 = (x-1)(x^2 + ax -2)

    x^3 - x^2 - 2x + 2 = x^3 +x^2(a-1) +x(-2-a) +2

    Now the coefficients of like powers of x on both sides of this equation
    must be equal, so in particular the coeficients of the linear terms are equal
    so:

    -2 = -2-a

    and so a=0.

    RonL
    I'll print this out and study it. Thank you so much for the detailed explanation.
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