help with a maths question (1 Viewer)

Sxmm

New Member
Joined
May 13, 2020
Messages
26
Gender
Male
HSC
2022
Let P(x) = (x+1)(x-3)Q(x) + a(x+1) + b, where Q(x) is a polynomial and a and b are real numbers

When P(x) is divisible by (x+1) the remainder is -11.
When P(x) is divisible by (x-3) the remainder is 1.

i) What is the value of b?

ii) What is the remainder when P(x) is divided by (x+1)(x-3)?

Would much appreciate it if anyone could give me a hand, thanks.
 

CM_Tutor

Moderator
Moderator
Joined
Mar 11, 2004
Messages
2,644
Gender
Male
HSC
N/A
The remainder theorem tells that the remainder when a polynomial is divided by is .

Applying this to the statement that "When P(x) is divisible by (x+1) the remainder is -11" tells us that:

We have divided by

Thus, the remainder is which, the question tells us, is , and so


Applying the same technique to the statement about dividing by should yield the value of .

Now, the division algorithm tells us that if I divide a polynomial by another polynomial (the divisor), where is of the same or lesser degree than , then the result will be a quotient, , and a remainder, . The quotient will have a degree that is no greater than the difference in the degree of and and the remainder will have a degree no higher than one less than that of . This gives us the formula


Using a concrete example, suppose I have a polynomial of degree 5 (say) and I divide it by , then I will get a quotient of degree no more than 4 and a remainder of degree no more than 0 (as the degree of is 1). In other words, will be a constant (, say).


This shows why the remainder theorem works, and also the factor theorem (if is a root of then is a factor, and thus .

Now, if I divide a polynomial by a divisor that is of degree 2 - something like , or , or ), then the remainder will be of at most degree 1, and so be in the form .


and this is the situation you have here. On division of by , the quotient will be and the remainder will be the degree 1 polynomial (or it could be a degree 0 polynomial, depending on the value of ), which you can simplify easily once you have found the values of and .

I get the remainder as , FYI.
 

Sxmm

New Member
Joined
May 13, 2020
Messages
26
Gender
Male
HSC
2022
The remainder theorem tells that the remainder when a polynomial is divided by is .

Applying this to the statement that "When P(x) is divisible by (x+1) the remainder is -11" tells us that:

We have divided by

Thus, the remainder is which, the question tells us, is , and so


Applying the same technique to the statement about dividing by should yield the value of .

Now, the division algorithm tells us that if I divide a polynomial by another polynomial (the divisor), where is of the same or lesser degree than , then the result will be a quotient, , and a remainder, . The quotient will have a degree that is no greater than the difference in the degree of and and the remainder will have a degree no higher than one less than that of . This gives us the formula


Using a concrete example, suppose I have a polynomial of degree 5 (say) and I divide it by , then I will get a quotient of degree no more than 4 and a remainder of degree no more than 0 (as the degree of is 1). In other words, will be a constant (, say).


This shows why the remainder theorem works, and also the factor theorem (if is a root of then is a factor, and thus .

Now, if I divide a polynomial by a divisor that is of degree 2 - something like , or , or ), then the remainder will be of at most degree 1, and so be in the form .


and this is the situation you have here. On division of by , the quotient will be and the remainder will be the degree 1 polynomial (or it could be a degree 0 polynomial, depending on the value of ), which you can simplify easily once you have found the values of and .

I get the remainder as , FYI.
thank you very much
 

Users Who Are Viewing This Thread (Users: 0, Guests: 1)

Top