Polynomial Roots
Factor Theorem
The polynomial P(x) is divisible by a polynomial of the form (x − a) if and only if P(x = a) = 0.
The value x = a is called the root or zero of P(x).
Roots of a Polynomial
These are the values to nullify the polynomial.
Calculate the Roots of the Polynomial:
P(x) = x2 − 5x + 6
P(2) = 22 − 5 · 2 + 6 = 4 − 10 + 6 = 0
P(3) = 32 − 5 · 3 + 6 = 9 − 15 + 6 = 0
x = 2 and x = 3 are roots or zeros of the polynomial: P(x) = x2 − 5x + 6, because P(2) = 0 and P(3) = 0.
Properties of the Roots and Factors of a Polynomial
1The zeros or roots are divisors of the independent term of the polynomial.
2For each root type x = a corresponds to it by a binomial of the type (x − a).
3 A polynomial can be expressed in factors by writing it as a product of all the binomials of type (x − a), which will correspond to the roots, x = a.
x2 − 5x + 6 = (x − 2) · (x − 3)
4The sum of the exponents of the binomial must be equal to the degree of the polynomial.
5All polynomials that do not have an independent term accept x = 0 as a root.
x2 + x = x · (x + 1)
Roots: x = 0, and x = − 1
6A polynomial is called irreducible or prime when it cannot be decomposed into factors.
P(x) = x2 + x + 1
Find the Roots and Factor the Following Polynomial:
Q(x) = x2 + x + 1
The divisors of the independent term are: ±1, ±2, ±3.
Q(1) = 12 − 1 − 6 ≠ 0
Q(−1) = (−1)2 − (−1) − 6 ≠ 0
Q(2) = 22 − 2 − 6 ≠ 0
Q(−2) = (−2)2 − (−2) − 6 = 4 +2 − 6 = 0
Q(3) = 32 − 3 − 6 = 9 − 3 − 6 = 0
The roots are: x = −2 and x = 3.
Q(x) = (x + 2) · (x − 3)
