How do I solve a polynomial with a missing term?

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Discussion Overview

The discussion revolves around solving the polynomial equation x^3 - 9x - 440 = 0, particularly focusing on finding the roots of the equation, including the known root x = 8. Participants explore various methods for solving the polynomial, including the rational root theorem and factoring techniques.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • Some participants suggest using the rational root theorem to identify potential integer roots of the polynomial, noting that any integer root must be a factor of 440.
  • Others propose that since the sign of the polynomial changes, there is at least one positive root, leading to the conclusion that x must be greater than 7.
  • One participant describes a method of rewriting the polynomial to facilitate factoring, ultimately leading to the identification of x = 8 as a root.
  • Another participant elaborates on the factoring process, demonstrating how to factor the polynomial into (x - 8)(x^2 + 8x + 55) and apply the quadratic formula to find the remaining roots.
  • Some participants express curiosity about the reasoning behind the choice of methods used to solve the polynomial, particularly regarding the use of the rational root theorem and factoring.

Areas of Agreement / Disagreement

Participants generally agree on the methods to approach the problem, such as using the rational root theorem and factoring. However, there are variations in the details of the approaches and the interpretations of the steps involved, indicating that multiple perspectives exist without a clear consensus on the best method.

Contextual Notes

Some participants' approaches depend on specific assumptions about the polynomial's structure and the applicability of the rational root theorem. The discussion includes various mathematical steps that may not be fully resolved or universally accepted.

blinky
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Hi,

I'm trying to help a high-school sophomore with a math problem, and unfortunately my algebra days are long behind me. Here's the equation:

x^3-9x-440=0

I know x=8, but I don't know how to find it. I'd appreciate some guidance.

Thanks.
 
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blinky said:
Hi,

I'm trying to help a high-school sophomore with a math problem, and unfortunately my algebra days are long behind me. Here's the equation:

x^3-9x-440=0

I know x=8, but I don't know how to find it. I'd appreciate some guidance.

Thanks.

use the rational root theorem. if x an integer is a zero it is factor of 440

so look at them 1,2,4,5,8,10,11,20,22,40,44,55,88,110,220,440 ( both plus and minus)

as the sign changes once so there is a positive root
x^3 = 440 + 9x or x^3 > 440 or x > 7
we need to check for values > 7( from the above set)
try 8 and it succeed
so we need not try further
you can divide by x-8 and get a quadratic and solve the same
 
kaliprasad said:
use the rational root theorem. if x an integer is a zero it is factor of 440

so look at them 1,2,4,5,8,10,11,20,22,40,44,55,88,110,220,440 ( both plus and minus)

as the sign changes once so there is a positive root
x^3 = 440 + 9x or x^3 > 440 or x > 7
we need to check for values > 7( from the above set)
try 8 and it succeed
so we need not try further
you can divide by x-8 and get a quadratic and solve the same

Thanks very much. I'll tell her.

Did you know just from looking at the equation that it had to be solved that way? If so, how did you know?
 
blinky said:
Thanks very much. I'll tell her.

Did you know just from looking at the equation that it had to be solved that way? If so, how did you know?

factoring is the way out. but if you cannot factor obviously, then we can use the rational root theorem to find the factor. This is one of the ways
 
kaliprasad said:
factoring is the way out. but if you cannot factor obviously, then we can use the rational root theorem to find the factor. This is one of the ways

Thanks again.
 
blinky said:
Hi,

I'm trying to help a high-school sophomore with a math problem, and unfortunately my algebra days are long behind me. Here's the equation:

x^3-9x-440=0

I know x=8, but I don't know how to find it. I'd appreciate some guidance.

Thanks.

Knowing that $x=8$, we could them proceed to rewrite the equation as:

$$x^3-8x^2+8x^2-64x+55x-440=0$$

Factor:

$$x^2(x-8)+8x(x-8)+55(x-8)=0$$

$$(x-8)\left(x^2+8x+55\right)=0$$

To get the other 2 roots, we apply the quadratic formula to the quadratic factor:

$$x=\frac{-8\pm\sqrt{8^2-4(1)(55)}}{2(1)}=-4\pm\sqrt{39}i$$
 
$x^3 - 9x -440$
$=x^3 - 8x^2 + 8x^2 - 64x + 55x - 440$
$=x^2(x - 8) + 8x(x - 8) + 55(x - 8)$
$=(x - 8)(x^2 + 8x + 55)$
$=(x - 8)(x^2 + 8x + 16 -16 + 55)$
$=(x - 8)((x + 4)^2 + 39)$
$=(x - 8)(x + 4)^2 - (-39))$
$=(x - 8)(x + 4)^2 - (\sqrt{39}i)^2$
$=(x - 8)(x + 4 - \sqrt{39}i)(x + 4 + \sqrt{39}i)$
 

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