Given f(x) = x^3 + 4x -5, show x-1 is a factor

  • Thread starter DeanBH
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  • #1
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The polynomial f(x) is given by f(x) = x^3 + 4x -5

show x-1 is factor :

f(1)=0
1+4-5=0 Thus it's a factor.

express in form (x-1)(x^2+px+q)

ends up being X^2 + X + 5.

next question is :

hence show the equation f(x) = 0 has one real root state it's value.

i know i have to use b^2 -4ac. If i do that it's -19 which is <0 so it's 1 value. But how do I find the value. And why does all this stuff work.

My teacher is one that just tells you how to do it. I have no idea why i'm doing it and it makes no sense so it's harder to remember. I would get 1or 2 marks out of 3 for this.

Firstly i dont know how to find out what the real number is. and i don't know why this works.

explain plez?! 8D
 

Answers and Replies

  • #2
HallsofIvy
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??? Didn't you start off by showing that f(1)= 0??

And you are trying x so that f(x)= 0??
 
  • #3
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First a short lesson in discriminants, since you asked why you use them.

Remember the equation for the roots of any polynomial [tex]f \left(x \right) = a x^{2} + bx + c [/tex] is
[tex] x = \frac{-b + \sqrt{b^{2} - 4 a c}}{2a}[/tex] and [tex] x = \frac{-b - \sqrt{b^{2} - 4 a c}}{2a}[/tex]

The term [tex]b^{2} - 4 a c[/tex] is called the "discriminant". If the discriminant is negative, then in you would be taking the square root of a negative number, which you cannot do with the real numbers.


Now, how many roots must [tex]f \left(x \right) = x^{3} + 4 x - 5[/tex] have?
Do you know the value of any of the roots already?
 
  • #4
tiny-tim
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Firstly i dont know how to find out what the real number is …
Hi DeanBH! :smile:

I think you're getting confused … the real number is 1, from the (x - 1) factor that you've already found.

When the question says "show the equation f(x) = 0 has one real root", it means exactly one real root (and two complex roots), instead of three real roots. That's all! :smile:
 
  • #5
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by working backwards, you can just show that x^3+4x-5 factors into (x^2+5)(x-1)

not sure if your teacher wants you to use the quadratic formula or not, but working backwards works just fine in some cases.
 
  • #6
HallsofIvy
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No, it doesn't.

(x2+ 5(x-1)= x3- x2+ 5x- 5, NOT x3+ 4x- 5. Did you mean (x2+ x+ 5)(x- 1)?
 
  • #7
i don't think it's is a difficult question ? .
 
  • #8
HallsofIvy
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I believe that's already been pointed out several times!
 
  • #9
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No, it doesn't.

(x2+ 5(x-1)= x3- x2+ 5x- 5, NOT x3+ 4x- 5. Did you mean (x2+ x+ 5)(x- 1)?

gah my bad, yes i ment (x2+ x+ 5)(x- 1)
 

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