Finding possible and actual rational roots

In summary, a rational root is a number expressed as a ratio of two integers, and can be found using the rational root theorem or other methods. Possible rational roots are potential solutions, while actual rational roots are the solutions when plugged in. A root is rational if it can be expressed as a ratio of two integers, and it can be determined by checking if the decimal form terminates or repeats. A polynomial can have no rational roots, but will have at least one complex root.
  • #1
duki
264
0

Homework Statement



a) 2x^4 - 15x^3 + 23x^2 + 15x - 25 = 0
b) 12x^3 - 20x^2 + 23x - 10 = 0

Homework Equations



The Attempt at a Solution



I was wondering if you guys could check my answers.

For A:

Possibles: +/- {1, 5, 25} / {1, 2}
Actual: 5, 5/2
 
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  • #2
And for B:

Possibles: +/- {1, 5, 10} / {1, 2, 3, 4, 6}
Actual: None
 
  • #3
duki said:
And for B:

Possibles: +/- {1, 5, 10} / {1, 2, 3, 4, 6}
Actual: None

Possibles: +/- {1, 2, 5, 10} / {1, 2, 3, 4, 6, 12}
 

1. What is the definition of a rational root?

A rational root is a number that can be expressed as a ratio of two integers, where the denominator is not equal to zero. In other words, it is a number that can be written in the form of a/b, where a and b are integers.

2. How do I find possible rational roots?

To find possible rational roots, you can use the rational root theorem. This theorem states that if a polynomial has integer coefficients, then any rational root must be of the form p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. You can also use synthetic division or a graphing calculator to find possible rational roots.

3. What is the difference between possible and actual rational roots?

Possible rational roots are the numbers that could potentially be solutions to a polynomial equation, while actual rational roots are the numbers that are actually solutions to the equation when plugged in. Possible rational roots may or may not be actual rational roots.

4. How do I know if a root is rational or irrational?

A rational root is a number that can be expressed as a ratio of two integers, while an irrational root is a number that cannot be expressed as a ratio of two integers. To determine if a root is rational or irrational, you can use a calculator or long division to see if the decimal form terminates or repeats.

5. Can a polynomial have no rational roots?

Yes, a polynomial can have no rational roots. For example, the polynomial x2 + 1 has no rational roots since there is no rational number that, when squared, equals -1. However, every polynomial with real coefficients will have at least one complex root, which may or may not be rational.

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