A root or zero of a function is a number that, when plugged in for the variable, makes the function equal to zero. Thus, the roots of a polynomial P(x) are values of x such that P(x) = 0 .
The Rational Zeros Theorem states:
If P(x) is a polynomial with integer coefficients and if is a zero of P(x) ( P() = 0 ), then p is a factor of the constant term of P(x) and q is a factor of the leading coefficient of P(x) .
We can use the Rational Zeros Theorem to find all the rational zeros of a polynomial. Here are the steps:
Example: Find all the rational zeros of
P(x) = x
^{3} -9x + 9 + 2x
^{4} -19x
^{2}
.
We can often use the rational zeros theorem to factor a polynomial. Using synthetic division, we can find one real root a and we can find the quotient when P(x) is divided by x - a . Next, we can use synthetic division to find one factor of the quotient. We can continue this process until the polynomial has been completely factored.
Example (as above): Factor
P(x) = 2x
^{4} + x
^{3} -19x
^{2} - 9x + 9
.
As seen from the second synthetic division above,
2x
^{4} + x
^{3} -19x
^{2} -9x + 9÷x + 1 = 2x
^{3} - x
^{2} - 18x + 9
. Thus,
P(x) = (x + 1)(2x
^{3} - x
^{2} - 18x + 9)
. The second term can be divided synthetically by
x + 3
to yield
2x
^{2} - 7x + 3
. Thus,
P(x) = (x + 1)(x + 3)(2x
^{2} - 7x + 3)
. The trinomial can then
be factored into
(x - 3)(2x - 1)
. Thus,
P(x) = (x + 1)(x + 3)(x - 3)(2x - 1)
. We can see that this solution is correct because the four rational roots
found above are zeros of our result.