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Polynomial devision and deiciding on the expression of the remainder. watch

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    The c3 book says things like "As the divisor is a quadratic and F(x) has the power of 4 then Q(x)must be a quadratic and the remainder must be a linear expression".

    How do you work out the outlines of the equations like this without actualy dividing F(x) by the divisor as the book did.
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    (Original post by Core)
    The c3 book says things like "As the divisor is a quadratic and F(x) has the power of 4 then Q(x)must be a quadratic and the remainder must be a linear expression".

    How do you work out the outlines of the equations like this without actualy dividing F(x) by the divisor as the book did.
    It's not entirely accurate.

    "As the divisor is a quadratic and F(x) has the power of 4 then Q(x)must be a quadratic and the remainder must be at most a linear expression".

    Highest power of F(x) [4] - hightest power of divisor [2] = power of Q(x) [2].

    You can see this as in the first part of the division process you look at the hightest power of the divisior and see what you need to multiply it by to get the hightest power of F(X), and when you multiply two numbers, or two x's you add the indices.

    The remainder must have a lower highest power than the divisor, otherwise you'd be able to carry on the division. In this case as the divisor has degree 2, the remainder has at most degree 1, i.e. linear.

    When doing the division the term in "x" may cancel and you are left with a constant, which is why I say "at most".
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    Intuitively, if you are dividing a degree 4 polynomial by a quadratic (degree 2), then q(x) will have a term of the form \frac{ax^4}{bx^2} = \frac{a}{b}x^2, which is the highest possible power in q(x).

    As for the remainder bit, think about integer division. If \frac{a}{b} = qb + r, i.e. with remainder r, and r\geq b, then the divison is incomplete as you can factor out another b and get \frac{a}{b} = q(b+1) + (r-b). The same principle applies for polynomials. If the remainder is a quadratic or a polynomial with a higher power, then you can factor out the divisor to get a linear remainder.
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    Thanks you both helped m alot.
 
 
 

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