# poly rational zero test by kp0vUawa

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```									Polynomials: The Rational Zero Test

The Rational Zero Test states that if a polynomial with real
coefficients has rational zeros each will be on the
following list:  c/d, where c is a factor of the constant
term and d is a factor of the leading coefficient.
Example: Make a list of possible rational zeros of the
polynomial, P(x) = 4x3 + 19x2 + 20x + 6.
The constant term is 6. Its factors are 1, 2, 3 and 6.
These are values of c.
The leading coefficient is 4. Its factors are 1, 2, and 4.
These are values of d.

Polynomials: The Rational Zero Test

c-values – 1, 2, 3 and 6              d-values – 1, 2 and 4

Now form all fractions of the form:  c/d:

1 2 3 6 1 2 3 6 1 2 3 6
 , , , , , , , , , , , .
1 1 1 1 2 2 2 2 4 4 4 4

Simplifying and removing duplicate fractions results in:
1 3 1       3
 1,  2,  3,  6,  ,  ,  ,  .
2 2 4       4

Polynomials: The Rational Zero Test

The polynomial in the example,
(P(x) = 4x3 + 19x2 + 20x + 6) actually has three real
3
zeros:  ,  2  2,  2  2.
4
Two of the zeros are irrational; only one is rational (- 3/4).
The Rational Zero Test makes no claims beyond those for
rational zeros. It simply states that if any rational zeros
exist, they will appear on the computed list. Indeed, - 3/4
was on the computed list!

Polynomials: The Rational Zero Test

Try: Make a list of possible rational zeros of the
polynomial, P(x) = 5x3 – 14x2 + 3x + 4.

1 2 4
The possible rational zeros are:  1,  2,  4,  ,  ,  .
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