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Can someone give the reason why please (dividing a polynomial by a quadratic)

Hi there :smile:.

Screenshot 2021-08-02 at 12.05.15.jpg

I'm reading the attached question and I do not understand why we stop dividing because the degree of -12x +5 is smaller than the denominator in the question.

Could someone please explain why do we stop dividing when we get to the part where the degree is now smaller?

(I've not put the rest of the answer up. It jus shows the values of A, B and C).

Thanks
Reply 1
Avatar for mqb2766
mqb2766
21
Original post by jojo55
Hi there :smile:.

Screenshot 2021-08-02 at 12.05.15.jpg

I'm reading the attached question and I do not understand why we stop dividing because the degree of -12x +5 is smaller than the denominator in the question.

Could someone please explain why do we stop dividing when we get to the part where the degree is now smaller?

(I've not put the rest of the answer up. It jus shows the values of A, B and C).

Thanks

As divsion is the inverse of multiplication and the result of multiplying some polynomial with a quadratic must follow
constant * quadratic = quadratic
linear * quadratic = cubic
quadratic * quadratic = quartic
...
If the right hand side is linear or constant, there is no polynomial which can multiply a quadratic to generate it. Hence you have a linear (or constant or zero) remainder after division by a quadratic.
(edited 2 years ago)
Think of it in terms of numbers, not polynomials, it works much the same way.

When the numerator is greater than the denominator, such as 9/5, you can split the fraction into 5/5 + 4/5, which is basically 1 + 4/5. When the numerator is no longer greater than or equal to the denominator, the fraction cannot be simplified further.

Similarly, when the polynomial in the numerator is of lower degree than the one in the denominator, the fraction cannot be simplified any more (at least for the purposes of this exercise). This is why you end up with a linear factor in the numerator, and a quadratic factor in the denominator.

Hope this helped!
(edited 2 years ago)
Reply 3
Avatar for jojo55
jojo55
OP
11
Thanks a lot. I understand it now :biggrin:.

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