Remainder/factor theorem question

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Discussion Overview

The discussion revolves around the application of the remainder and factor theorems to a polynomial function defined as $$f\left(x\right)=x^{4}+4x^{3}-x^{2}-16x-12$$. Participants explore the division of this polynomial by specific linear factors and the implications of their findings regarding the existence of remainders and the identification of factors.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Homework-related
  • Debate/contested

Main Points Raised

  • Post 1 presents a question about showing that there is no remainder when dividing $$f\left(x\right)$$ by $$(x+1)$$ and using the factor theorem to show that $$(x+2)$$ is a factor of $$f\left(x\right)$$.
  • Post 1 claims that the remainder is 0 when dividing by $$(x+1)$$, but does not provide a complete justification for this assertion.
  • Post 1 calculates $$f(-2)$$ to demonstrate that $$(x+2)$$ is a factor, arriving at a conclusion that $$f(-2)=0$$.
  • Post 2 provides a detailed long division process, suggesting that the result leads to a factorization of the polynomial.
  • Post 3 repeats the long division steps and suggests that the remaining factors could be $$(x-2)(x+3)$$, but expresses confusion regarding the notation used in the division.
  • Post 4 confirms the factorization proposed in Post 3 and clarifies a misunderstanding about the notation used in the long division, emphasizing that it is not a square root.

Areas of Agreement / Disagreement

Participants generally agree on the steps taken in the long division process and the identification of factors, but there is some confusion regarding the notation and the interpretation of the results. The discussion remains somewhat unresolved regarding the clarity of the long division notation and its implications for the factorization.

Contextual Notes

There are limitations in the clarity of the long division notation, with some participants misinterpreting it as a square root. Additionally, the steps leading to the final factorization are not fully detailed, leaving some assumptions unaddressed.

ai93
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Question

A function $$f\left(x\right)$$ is defined by f\left(x\right)=x^{4}+4x^{3}-xx^{2}-16x-12
a) Show that there is no remainder when $$f\left(x\right)$$ is divided by $$(x+1)$$

b)Use the factor theorem to show that $$(x+2)$$ is a factor of $$f\left(x\right)$$

c) Using answers to a) and b) determine the remaining factors by the long division method.

MY SOLUTIONa) $$(x+1)\sqrt{x^{4}+4x^{3}-x^{2}-16x-12}$$ = $$x^{2}+3x^{2}-4x-12$$
Remainder = 0

b) If $$(x+2)$$ is a factor then $$f(-2)=0$$

$$\therefore f(-2)=(-2)^{4}+4(-2)^{3}-(-2)^{2}-16(-2)-12$$ = 0
So (x+2) is a factor.

c)
(x+1)(x+2)=$$x^{2}+3x+2$$

I tried to use long division

$$(x^{2}+3x+2)\sqrt{x^{4}+4x^{3}-xx^{2}-16x-12}$$

But I am having trouble finding the last remaining factors?
How to solve by long division?
 
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The indicated long division is carried out as follows:

$$\begin{array}{r}x^2+x-6\hspace{102px}\\x^2+3x+2\enclose{longdiv}{x^4+4x^3-x^2-16x-12} \\ -\underline{\left(x^4+3x^3+2x^2\right)} \hspace{62px} \\ x^3-3x^2-16x \hspace{38px} \\ -\underline{\left(x^3+3x^2+2x\right)} \hspace{35px} \\ -6x^2-18x-12 \\ -\underline{\left(-6x^2-18x-12\right)} \hspace{-10px} \\ 0 \end{array}$$

Now you just need to factor the dividend. :D
 
MarkFL said:
The indicated long division is carried out as follows:

$$\begin{array}{r}x^2+x-6\hspace{102px}\\x^2+3x+2\enclose{longdiv}{x^4+4x^3-x^2-16x-12} \\ -\underline{\left(x^4+3x^3+2x^2\right)} \hspace{62px} \\ x^3-3x^2-16x \hspace{38px} \\ -\underline{\left(x^3+3x^2+2x\right)} \hspace{35px} \\ -6x^2-18x-12 \\ -\underline{\left(-6x^2-18x-12\right)} \hspace{-10px} \\ 0 \end{array}$$

Now you just need to factor the dividend. :D

I had initially got that with my rough working out, but was confused as there are 5 terms in the square root!

So the other factors would be $$(x-2)(x+3)$$ :D
 
mathsheadache said:
I had initially got that with my rough working out, but was confused as there are 5 terms in the square root!

So the other factors would be $$(x-2)(x+3)$$ :D

Your factorization of the dividend is correct. That's not a square root though, that is the long division symbol, which denotes a very different operation. :D
 

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