The remainder and factor theorems

• Apr 7th 2009, 08:53 AM
Mr Rayon
The remainder and factor theorems
The remainder when x^2 - 3x + 1 is divided by (x + d) is 11. Find the possible values of d.
• Apr 7th 2009, 09:22 AM
stapel
Quote:

Originally Posted by Mr Rayon
The remainder when x^2 - 3x + 1 is divided by (x + d) is 11. Find the possible values of d.

Since the divisor is x + d, then the zero (when doing the synthetic division) is -d. So do the synthetic division:

Code:

-d | 1    -3        1   |      -d    d^2+3d   +------------------     1  -d-3  d^2+3d+1
Set the remainder equal to the given value:

. . . . . $d^2\, +\, 3d\, +\, 1\, =\, 11$

This rearranges as:

. . . . . $d^2\, +\, 3d\, -\, 10\, =\, 0$

Either factor the quadratic and then solve the factors for the values of d, or else plug the above into the Quadratic Formula.

(Wink)
• Apr 7th 2009, 09:58 AM
Soroban
Hello, Mr Rayon!

Quote:

The remainder when $x^2 - 3x + 1$ is divided by $(x + d)$ is 11.
Find the possible values of $d.$

From the heading, I assume you're expected to know the Remainder Theorem.

. . If a polynomial $p(x)$ is divided by $(x-a)$, the remainder is $p(a).$

We have: . $p(x) \:=\:x^2-3x+1$

When $p(x)$ is divided by $(x+d)$, the remainder is $p(\text{-}d)$

We have: . $p(x) \:=\:x^2-3x+1$ divided by $(x-[\text{-}d])$

Hence: . $(\text{-}d)^2 - 3(\text{-}d) + 1 \:=\:1 \quad\Rightarrow\quad d^2 + 3d \:=\:0 \quad\Rightarrow\quad d(d+3) \:=\:0$

Therefore: . $d \;=\;0,\:\text{-}3$

• Apr 7th 2009, 05:15 PM
Mr Rayon
Quote:

Originally Posted by Soroban
Hence: . $(\text{-}d)^2 - 3(\text{-}d) + 1 \:=\:1$

...But what happened to the 11?
• Apr 7th 2009, 05:29 PM
mr fantastic
Quote:

Originally Posted by Mr Rayon
...But what happened to the 11?

A small mistake by Soroban is what happened. But surely you can make the appropriate corrections .... Not that you need to do this, given Stapel's earlier post. Did you read it by the way?
• Apr 7th 2009, 05:40 PM
Mr Rayon
Quote:

Originally Posted by mr fantastic
A small mistake by Soroban is what happened. But surely you can make the appropriate corrections .... Not that you need to do this, given Stapel's earlier post. Did you read it by the way?

Yes...I never skip a post

d^2 + 3d - 10 = 0

Using the quadratic formula we get:

(-3 - 7)/2 = -5 or (-3 +7)/2 = 2

But are there any other ways to solve this apart from by factorising?
• Apr 7th 2009, 05:41 PM
stapel
Quote:

Originally Posted by Mr Rayon
But are there any other ways to solve this apart from by factorising?

Yes; you can solve the quadratic by using the Quadratic Formula, or you can complete the square. :D
• Apr 7th 2009, 05:45 PM
Mr Rayon
Quote:

Originally Posted by stapel
Yes; you can solve the quadratic by using the Quadratic Formula, or you can complete the square. :D

Yes...I tried to use completing the square but it got a bit messy down the end. I prefer using the quadratic formula.