How can you deduce a cubic polynomial given

Leroy Davidson

Leroy Davidson

Answered question

2022-03-25

How can you deduce a cubic polynomial given two quadratic divisors, their respective remainders, and nothing else?
A cubic polynomial gives the remainders (5x+4) and (12x1) when divided by (x2x+2) and (x2+x1) respectively.
How can I find the polynomial from this set of information?

Answer & Explanation

Denise Daniel

Denise Daniel

Beginner2022-03-26Added 8 answers

Let p(x) be the unknown polynomial.
Then for some unknown constants a, b, c, d, we must have the identities

(1) p ( x ) = ( a x + b ) ( x 2 x + 2 ) + ( 5 x + 4 ) (2) p ( x ) = ( c x + d ) ( x 2 + x 1 ) + ( 12 x 1 )

Hence identically we have
(ax+b)(x2x+2)+(5x+4)=(cx+d)(x2+x1)+(12x1)
or equivalent,
ax3+(ba)x2+(2ab+5)x+(2b+4)=cx3+(c+d)x2+(dc+12)x(d+1)
which yields the system
a=cba=c+d2ab+5=dc+122b+4=d1
of 4 linear equations in 4 unknowns.
To finish the task, solve the system and then substitute the results in either (1) or (2) to find p(x).

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