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.
- 5
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+5x^2+x + 5
We can check for integer and rational roots one at a time.
Checking for Integer Roots
The constant term of this polynomial is 5. Its factors, and the possible integer roots for P(x)=0, are ± 1 and ± 5. Let's check them! Remember, a root will give us a result of 0 after substituted into the polynomial.
x
x^3+5x^2+x+5
P(x)=x^3+5x^2+x+5
1
1^3+5( 1)^2+ 1+5
12 *
- 1
( - 1)^3+5( - 1)^2+( - 1)+5
8 *
5
5^3+5( 5)^2+ 5+5
260 *
- 5
( - 5)^3+5( - 5)^2+( - 5)+5
0 âś“
Looking at the table, we can see that - 5 is a root for P(x)=0.
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 - 5. Since the leading coefficient is 1, the possible rational roots are all integer numbers, which were already considered in the prior table.
