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.
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)= 3x^4-4x^3-x^2 - 7
We can check for integer and rational roots one at a time.
Checking for Integer Roots
The constant term of this polynomial is - 7. Its factors, and the possible integer roots for P(x)=0, are ± 1 and ± 7.
Checking for Non-integer Roots
Next, let's try to find non-integer rational roots. The leading coefficient is 3 and the constant term is - 7. Therefore, the possible non-integer roots are ± 1 3 and ± 7 3.
The possible rational roots of P(x)=0 include ± 1, ± 7, ± 13, and ± 73.
