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)