Polynomials(factor and remainder theorum) (1 Viewer)

[The Dave]

Hey guys i was just wondering what the factor and remainder theorum state and exactly how they work, i remember roughly what they do but can't seem to find the notes i worte in class ages ago. ANy help would be much apprecaited? thanks guys

 

[withoutaface]

Factor theorem states that if P(a)=0, then (x-a) is a factor, and vice versa.
This works because, if Q(x) has only positive powers of x

P(x)/(x-a)=Q(x)
P(x)=(x-a)Q(x)
P(a)=0*Q(x)

Remainder theorem states that if P(b)=c, then the remainder when P(x) is divided by (x-b) is c.

This works because P(x)/(x-b)=Q(x)+[R(x)/(x-b)]
P(x)=(x-b)Q(x)+R(x)
P(b)=0*Q(b) +R(b)
c=R(b)

Hopefully that helped

 

[coca cola]

Remainder theorem states that if P(b)=c, then the remainder when P(x) is divided by (x-b) is c.

errr i think there is something wrong with your statement.

it should be when P(x) is divided by (x-b) this produces Q(x)+[R(x)/(x-b)].

 

[coca cola]

just a tip with polynomials. the degree or any other power is always a positive interger. it can't be a fraction or a negative.

when we divide a polynomial say P(x) by another polynomial A(x), we want a polynomial solution, this is why reminders is produced when the A(x) is not a factor of P(x).

think about long division you did in primary school. we want a interger solution, hence if it does not factorise we create something called a remainder with the bit that does factorise.

 

[shazzam]

the degree or any other power is always a positive interger. it can't be a fraction or a negative.

But a polynomial with degree zero is a constant. This comes into relevance in certain questions

 

[The Dave]

THanks guys, much appreciated. It's all coming back to me now after reading this, thanks.