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.