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# Remainder Theorem Question

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1. 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.
2. (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.
3. (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 by you would be left with an expression in the form and is the remainder which is an integer.

For the remainder theorem, however, you simply plug through the cubic and the result is the remainder; which is obviously an integer.
4. (Original post by RDKGames)
If you were to divide by you would be left with an expression in the form and is the remainder which is an integer.

For the remainder theorem, however, you simply plug 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 by

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
5. (Original post by FamilyFirst)
I see.

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

Divide by

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 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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