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Math Help - Polynomial function

  1. #1
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    Polynomial function

    If f(x)=x^4+2x^3+5x^2-16x-20 , show that f(x) can be expressed in the form (x^2+x+a)^2-4(x+b)^2 .

    I don really know how to show , but i do know how to determine the value of a and b . Is there a way ?
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  2. #2
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    Quote Originally Posted by mathaddict View Post
    If f(x)=x^4+2x^3+5x^2-16x-20 , show that f(x) can be expressed in the form (x^2+x+a)^2-4(x+b)^2 .

    I don really know how to show , but i do know how to determine the value of a and b . Is there a way ?
    Expand the second one and then equate the coefficients of like powers of x.
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  3. #3
    Senior Member MacstersUndead's Avatar
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    If what your first post says is correct in your saying that you have determined values of such a and b, you have proved the statement.
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    Re :

    Quote Originally Posted by MacstersUndead View Post
    If what your first post says is correct in your saying that you have determined values of such a and b, you have proved the statement.

    Did u mean that by just finding the values of a and b is sufficient to prove that statement ?
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  5. #5
    Like a stone-audioslave ADARSH's Avatar
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    Quote Originally Posted by mathaddict View Post
    If f(x)=x^4+2x^3+5x^2-16x-20 , show that f(x) can be expressed in the form (x^2+x+a)^2-4(x+b)^2 .

    I don really know how to show , but i do know how to determine the value of a and b . Is there a way ?
    (x^2+x+a)^2-4(x+b)^2 .

    =x^4+x^2 + a^2 + 2x^3 + 2ax + 2ax^2 -4x^2-4b^2 -8xb

     <br />
=x^4 + 2x^3+(2a-3)x^2 + (2a-8b)x +(a^2 -4b^2)<br />

    Comparing above eqn. with

    f(x)=x^4+2x^3+5x^2-16x-20

     <br />
2a-3 = 5<br />

     <br />
\implies a=4<br />

     <br />
2a-8b= -16<br />

     <br />
\implies b= 3<br />

    You can check by putting the value
     <br />
a^2-4b^2 = -20<br />
    This is what you did

    Hence you have shown that values of a and b are real , proving that statement is absolutely correct
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