Polynomials and the Factor Theorum... (1 Viewer)

[GaDaMIt]

From Cambridge 3u Year 12 book...

The polynomial P(x) is divided by (x-1)(x+2). Find the remainder, given that P(1) = 2 and P(-2) = 5. [HINT: The remainder may have degree 1.]


dunno how to approach this question.. any help will be greatly appreciated

 

[SoulSearcher]

Ok, P(x) = (x-1)(x+2)Q(x) + R(x), where R(x) is the remainder
Let R(x) = ax + b, so that
P(x) = (x-1)(x+2)Q(x) + (ax+b)
From here, you sub in the values x = 1 and x = -2, so that you get 2 equations:
P(1) = a + b
P(-2) = b - 2a
But you are given that P(1) = 2 and P(-2) = 5, therefore you get the equations:
2 = a + b
5 = b - 2a
Now just simultaneously solve them to find a and b, and then you sub the values of a and b into R(x) to find the remainder.

 

[Riviet]

p(x)=A(x).(x-1)(x+2) + R(x) where R(x) is the remainder.

Since (x-1)(x+2) is of degree 2, the remainder must be less than degree 2, ie it could be of degree 0 (constant) or 1 (linear), let this remainder be ax+b.

So p(x)=A(x).(x-1)(x+2) + (ax+b)

Given p(1)=2,
p(1)=A(1).(1-1)(1+2) + (a+b)=2
p(1)=0+a+b=2

.'. a+b=2 (1)

Given p(-2)=5,
p(-2)=A(-2).(-2-1)(-2+2) + (-2a+b)=5
p(-2)=0-2a+b=5

.'. -2a+b=5 (2)

Solve (1) and (2) simultaneously to find a and b, which will form your remainder, ax+b.

- beaten by SoulSearcher.

 

[hyparzero]

From Cambridge 3u Year 12 book...

The polynomial P(x) is divided by (x-1)(x+2). Find the remainder, given that P(1) = 2 and P(-2) = 5. [HINT: The remainder may have degree 1.]


dunno how to approach this question.. any help will be greatly appreciated

wat exactly is the question....

 

[SoulSearcher]

- beaten by SoulSearcher.

Yeah, but then I never really explain how I do the question

 

[GaDaMIt]

another polynomial question ..

Find the monic, even quartic that has y intercept 9 and a zero at x=3



Nevermind

 

[GaDaMIt]

URGENT Question..

When a polynomial is divided by (2x + 1)(x - 3), the remainder is 3x - 1. What is the remainder when the polynomial is divided by 2x + 1


The trouble im having is that cambridge answers contradict my own, and i havent found a mistake in them before, so im not entirely sure im doing it correct..

I get -5/2 .. answers get 1/2

 

[Riviet]

According to the question posted, you appear to be correct. What's interesting is if you change the quadratic being divided to (2x-1)(x-3), you will get Cambridge's answer. So I reckon it's either a typo in either the answer or the question.