Polynomial Question. (1 Viewer)

cheezz

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A monic cubic polynomial when divided by x&shy;<SUP>2</SUP> + 4 leaves a remainder of x + 8 and when divided by x leaves a remainder of -4. Find the polynomial in the form ax<SUP>3</SUP> + bx<SUP>2</SUP> + cx + d.

I know there is a simple way to do this but i cant see it.
 

untouchablecuz

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A monic cubic polynomial when divided by x*<sup>2</sup> + 4 leaves a remainder of x + 8 and when divided by x leaves a remainder of -4. Find the polynomial in the form ax<sup>3</sup> + bx<sup>2</sup> + cx + d.

I know there is a simple way to do this but i cant see it.
let P(x) be the polynomial P(x)=ax2+bx2+cx+d=x2+bx2+cx+d, since P(x) is monic

by the division transformation,

P(x)/D(x)≡Q(x)+R(X)/D(x)

here, D(x)=x2+4 and R(x)=x+8

so, P(x)≡(x2+4)Q(x)+(x+8)

because P(x) is cubic and monic, Q(x) is linear i.e. Q(x)≡(x-k)

so P(x)≡(x2+4)Q(x-k)+(x+8)

the remainder when P(x) is divided by (x-0) is P(0)=R(0)

but R(0)=P(0)=-4

.'. P(0)=(02+4)(0-k)+(0+8)=-4 → k=3

hence P(x)=(x2+4)(x-3)+(x+8)=x3-3x2+4x-12+(x+8)=x3-3x2+5x-4
 
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cheezz

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okay... thanks... i get everything except the condition of monic and cubic giving rise to Q(x) being the same as (x-k).

can you explain that part? we rushed through all this in class.

cheers.
 
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untouchablecuz

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basically what monic means is that the co-efficient of the term of the highest power is 1

now, polynomial theory states that if a polynomial is divided by a divisor D(x), then the quotient Q(x) has a degree that is equal to degree(P(x))-degree(D(x))

in this case

degree(P(x))=3

degree(D(x))=2

so degree(Q(x))=1

if the degree is 1, that is, Q(x) is linear, then Q(x) will be of the form (mx-k)

P(x)=(x2+4)(mx-k)+(x+8)=(m)x3+...

we are given that P(x) is monic, so the co-efficient of the highest power (x3) will be 1

that is, m=1 and thus Q(x) is of the form (x-k)
 
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cheezz

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okay... so the picture is getting clearer and clearer haha.
but when you mean degree... do you mean highest power?
and you said quotient has a degree equal to degree(P(x)) - degree(Q(x))...
did you not mean degree(P(x)) - degree(D(x))?? because then everything makes sense.

thanks for your help eh... sorry for seeming like such a downie.
 

untouchablecuz

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okay... so the picture is getting clearer and clearer haha.
but when you mean degree... do you mean highest power?
essentially, yes; the degree of a polynomial is equal to the highest power

and you said quotient has a degree equal to degree(P(x)) - degree(Q(x))...
did you not mean degree(P(x)) - degree(D(x))?? because then everything makes sense.
yes thats what i meant, sorry

thanks for your help eh... sorry for seeming like such a downie.
no worries m8, good luck with HSC
 

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