Polynomial Question. (1 Viewer)

[cheezz]

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]

A monic cubic polynomial when divided by x*² + 4 leaves a remainder of x + 8 and when divided by x leaves a remainder of -4. Find the polynomial in the form ax³ + bx² + 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

 

[Drongoski]

P(x) = x³ -3x² + 5x - 4 ???

 

[untouchablecuz]

oo yes sorry, forgot to add the x+8 on

fixed

 

[cheezz]

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.

 

[untouchablecuz]

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)

 

[cheezz]

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]

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