If there are any integer roots, they must be factors of the constant term. If there are any rational roots, they have the form ± pq, where p is a factor of the constant term and q a factor of the leading coefficient.
± 1, ± 2, ± 3, ± 6
Practice makes perfect
One way of finding the roots for P(x)=0 is to guess and check. This is inefficient unless there is a way to minimize the number of possible roots. The Rational Root Theorem helps us with this!
Let Q(x)= a_nx^n+a_(n-1)x^(n-1)+... +a_1x+ a_0 be a polynomial with integer coefficients. There are a limited number of possible roots for Q(x)=0.
Rational roots must reduce to be p q, where p is an integer factor of a_0, and q is an integer factor of a_n.
Now let's consider the given polynomial.
P(x)= 1x^3+4x^2-10x + 6
We can check for possible integer and rational roots one at a time.
Checking for Integer Roots
The constant term of this polynomial is 6. Its factors, and the possible integer roots for P(x)=0, are ± 1, ± 2, ± 3, and ± 6.
Checking for Non-integer Roots
Next, let's try to find non-integer rational roots. The leading coefficient is 1 and the constant term is 6. Therefore, the possible rational roots are ± 1 1, ± 2 1, ± 3 1, and ± 6 1. Note that these all simplify to integer numbers, and they were already considered above.
The possible rational roots of P(x)=0 include ± 1, ± 2, ± 3, and ± 6.
