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I have been presented with the following question :

The polynomial

$$f(x) = x^3 - 2x^2 +ax + b$$

satisfies the following :

a) It is divisible (x-1)

b) it leaves a remainder of -24 when divided by (x+3)

I have no idea where to begin with this, I have tried long division but I get stuck when ax becomes the dividend. I have also tried substitution and rearanging the equation but im not getting anywhere. If anyone could show me how to complete this question, or shed any light on the steps I would be extremely grateful.

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So $f(x)$ is divisible by $(x-1)$ also can be restated as

$$f(x) = (x-1)*\text{something}$$

$$f(1) = (1-1)*\text{something} = 0*\text{something} = 0$$

So we have that

$$ f(1) = 1^3 - 2*1^2 +a*1 + b = a+b-1 = 0$$

Now the other condition is that $f(x)$ leaves remainder $-24$ when divided by $x+3$. That is to say $f(x) +24$ is divisible by $x+3$ meaning

$$ f(-3) + 24 = 0 \rightarrow (-3)^3-2(-3)^2-3a + b + 24 =0 \rightarrow $$

$$ -27-18-3a +b+24 =0 \rightarrow -21 -3a + b = 0$$

So now we have a system of two equations in two unknowns:

$$ a+b-1 = 0 \\ -21 -3a + b = 0 $$

We solve:

(constants on right side) $$ a+b = 1 \\ -3a + b = 21 $$

(adding top to bottom 3 times) $$ a +b = 1 \\ 4b = 24 \rightarrow b = 6 $$

(substituting $b$ into top) $$ a = -5$$

So $x^3 - 2x^2 - 5x + 6$ is your final answer.

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  • $\begingroup$ I made a silly mistake somewhere here. Disregard the later calcuations, i'm going to fix it $\endgroup$ – frogeyedpeas May 15 '16 at 22:57
  • $\begingroup$ got it! now go ahead and read it through, ask away for any parts that don't make sense $\endgroup$ – frogeyedpeas May 15 '16 at 23:00
  • $\begingroup$ Thank you very much for the breakdown on a step by step level, this makes sense. I need to go back and revise simultaneous equations, its been a while ! $\endgroup$ – Flewitt Connor May 15 '16 at 23:35
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What you're given is

$$(a)\;\; f(1)=0\;,\;\;\;(b)\;\;f(-3)=-24$$

Observe you get two equations for $\;a,b\;$ :

$$a+b=1\\-3a+b=21$$

Solve the easy system now.

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  • $\begingroup$ Thank you I see now how remainder -24 simply means = -24 instead of =0. this has certainly helped $\endgroup$ – Flewitt Connor May 15 '16 at 23:43
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Hint:

Every polynomial $f(x)$ divided by a a first degree polynomial of the form $x-\rho$ satisfies: $$f(x) = (x-\rho) \cdot \pi(x) + r,$$ where $r$ is the remainder of the division. Now, what is $f(-1) $ and what is $f(-3)$?

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