Solving for X in an X^4 Equation: Homework Help and Explanation

  • Thread starter Physicsissuef
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In summary, the conversation discusses finding the value of x in the given equation and suggests using the Rational Zeros Theorem to solve it. Some possible candidates for factoring and solutions are also mentioned. Ultimately, the conversation leads to finding four solutions for x and reducing the equation to a quadratic.
  • #1
Physicsissuef
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Homework Statement


Find x.

[tex]x^4+4x^3+3x^2-20x-20=0[/tex]

Homework Equations




The Attempt at a Solution



[tex]x^4+4x^3+3x^2-20x-20=0[/tex]/:[tex]x^2[/tex]

[tex]x^2+4x+3-20\frac{1}{x}-20\frac{1}{x^2}=0[/tex]

[tex](x^2-20\frac{1}{x^2})+4(x-5\frac{1}{x})+3=0[/tex]

[tex]x-5\frac{1}{x}=y[/tex]

[tex]x^2-10+25\frac{1}{x^2}=y^2[/tex]

[tex]x^2+25\frac{1}{x^2}=y^2+10[/tex]

? I can't substitute for [tex]x^2-20\frac{1}{x^2}[/tex]
 
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  • #2
Perhaps thehttp://www.sparknotes.com/math/algebra2/polynomials/section4.rhtml" may help you!
 
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  • #3
Sometimes it's a good idea to simply guess a solution.

Good candidates are integers and the rationals allowed by the theorem mentioned by colby.

Once you have found one solution (no matter how) you can factor out this root and are left with a polynomial of degree one less than the original one.

Can you guess any solution to your equation?

What about x=0 ? x=1 ? x=2 ? x= -1 ? x= -2 ?
 
  • #4
Is it a candidate for factoring into three factors? A quadradic and two functions of x ?
 
  • #5
jimvoit said:
Is it a candidate for factoring into three factors? A quadradic and two functions of x ?

Yes.
 
  • #6
Yes, this problem can be solved with Perhaps the Rational Zeros Theorem. thanks for all guys.
 
  • #7
The terms "-20x - 20" jumped out at me. What value of x will make those two terms cancel? Will that same value of x make the other terms cancel out?

That's one factor down, three to go.
 
  • #8
The next root is just as easy and then you are left with a quadratic!
 

1. What is the basic concept behind solving for X in an X^4 equation?

The basic concept behind solving for X in an X^4 equation is to find the value (or values) of X that make the equation true. This is done by manipulating the equation using algebraic techniques and properties, such as factoring, expanding, and isolating the variable.

2. How do I know if an X^4 equation has real solutions?

An X^4 equation will have real solutions if the discriminant (b^2 - 4ac) is greater than or equal to 0, where a, b, and c are the coefficients of the equation in standard form (ax^4 + bx^3 + cx^2 + dx + e = 0). If the discriminant is less than 0, the equation will have imaginary solutions.

3. Is there a specific order in which I should solve for X in an X^4 equation?

There is no specific order in which you must solve for X in an X^4 equation. However, it is often helpful to start by factoring out a common factor or grouping terms together to simplify the equation before attempting to solve for X.

4. Can I use the quadratic formula to solve an X^4 equation?

No, the quadratic formula (x = [-b ± √(b^2 - 4ac)]/2a) can only be used to solve quadratic equations (equations with an X^2 term). To solve an X^4 equation, you will need to use other algebraic techniques, such as factoring or the method of undetermined coefficients.

5. How do I check my solution for an X^4 equation?

To check your solution for an X^4 equation, you can substitute the value of X into the original equation and see if it satisfies the equation. You can also plug the value into a graphing calculator and see if it produces the correct graph or table of values.

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