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    In this screenshot: https://gyazo.com/5052812379fedee11586ea1db3d55b78

    It states "As the divisor is a linear expression and F(X) is a cubic polynomial then Q(X) must be a quadratic and the remainder must be a constant.I don't understand how they determined it would be a constant by that information.
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    (Original post by FamilyFirst)
    In this screenshot: https://gyazo.com/5052812379fedee11586ea1db3d55b78

    It states "As the divisor is a linear expression and F(X) is a cubic polynomial then Q(X) must be a quadratic and the remainder must be a constant.I don't understand how they determined it would be a constant by that information.
    By the remainder theorem, the remainder will be equal to the value of the cubic expression at x=3, and clearly that value is just a constant.
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    (Original post by FamilyFirst)
    In this screenshot: https://gyazo.com/5052812379fedee11586ea1db3d55b78

    It states "As the divisor is a linear expression and F(X) is a cubic polynomial then Q(X) must be a quadratic and the remainder must be a constant.I don't understand how they determined it would be a constant by that information.
    If you were to divide f(x) by x-3 you would be left with an expression in the form ax^2+bx+c+\frac{d}{x-3} and d is the remainder which is an integer.

    For the remainder theorem, however, you simply plug x=3 through the cubic and the result is the remainder; which is obviously an integer.
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    (Original post by RDKGames)
    If you were to divide f(x) by x-3 you would be left with an expression in the form ax^2+bx+c+\frac{d}{x-3} and d is the remainder which is an integer.

    For the remainder theorem, however, you simply plug x=3 through the cubic and the result is the remainder; which is obviously an integer.
    I see.

    Without using long-division how would you work out the form something like this would take on?

    Divide  x^4 + x^3 + x - 10 by  x^2 + 2x - 3

    The reason I'm asking is because in the book it states "As the divisor is a quadratic F(X) has a power of 4 then Q(x) must be a quadratic and the remainder must be a linear expression" This doesn't explain why though
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    (Original post by FamilyFirst)
    I see.

    Without using long-division how would you work out the form something like this would take on?

    Divide  x^4 + x^3 + x - 10 by  x^2 + 2x - 3

    The reason I'm asking is because in the book it states "As the divisor is a quadratic F(X) has a power of 4 then Q(x) must be a quadratic and the remainder must be a linear expression" This doesn't explain why though
    I would just use long-division; otherwise reverse grid method. I recommend you learn long-division.

    In that case the divided expression would be in the form Ax^2+Bx+C+\frac{Dx+E}{x^2+2x-3} and you cannot divide a linear expression by a quadratic one any further than that (referring to the last term); hence the remainder is a linear expression. If you follow through with long division you can observe this.
 
 
 
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