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)= 4x^4-2x^3+x^2 - 12
We can check for integer and rational roots one at a time.
Checking for Integer Roots
The constant term of this polynomial is - 12. Its factors, and the possible integer roots for P(x)=0, are ± 1, ± 2, ± 3, ± 4, ± 6, and ± 12.
Checking for Non-integer Roots
Next, let's try to find non-integer rational roots. The leading coefficient is 4 and the constant term is - 12. Therefore, the possible rational roots are ± 1 2, ± 2 2, ± 3 2, ± 4 2, ± 6 2, ± 12 2, ± 1 4, ± 2 4, ± 3 4, ± 4 4, ± 6 4, and ± 12 4. Note that ± 2 2 =± 4 4=± 1, ± 4 2=± 2, ± 6 2 =± 12 4=± 3, and ± 12 2=± 6 are integer numbers, and they were already considered when we checked for integer roots. Additionally, ± 6 4 simplifies to ± 3 2 and ± 2 4 simplifies to ± 1 2. Therefore, the possible non-integer roots include ± 14, ± 12, ± 34, and ± 32.
The possible rational roots for P(x)=0 include ±1, ±2, ±3, ±4, ±6, ±12, ± 14, ± 12, ± 34, and ± 32.
