Day 5 – The Factor Theorem
The Factor Theorem is a special case of the Remainder Theorem. It states that:
A polynomial P(x) has x – b as a factor if and only if P(b) = 0.
If a polynomial P(x) has x – b as a factor, then there is no remainder and
therefore:
P ( x )  ( x  b)  Q ( x )
P (b)  (b  b)  Q (b)
P (b )  0
The reverse is also true, that is if P(b) = 0, then x – b is a factor of P(x).
Note: This is also true when the factor is ax – b and a  1 .
Ex. Show that x – 2 is a factor of x3  3x 2  4 x  4 .
The remainder is zero, therefore x – 2 is a factor.
Try. State whether 2x + 3 is a factor of 6 x3  9 x 2  2 x  3 .
Integral Zero Theorem
The integral zero theorem can be used to find factors of polynomials if the
coefficient of the highest degree term is 1.
If x = b is an integral zero of a polynomial P(x) with integral coefficients, then
b is a factor of the constant term of the polynomial. (ie. x – b is a factor.)
Ex. Find the possible values of b (therefore the possible factors) for the polynomial
x3  3x 2  13x  15 . Find one value that is a factor.
Possible values of b: 1, 3, 5, 15 .
Use the factor theorem to determine one factor.
Choose a value of b to try.
What if we want to fully factor the polynomial?
Once we have found one factor we can use synthetic division (or long division).
The coefficients of the polynomial go on top, the value of b goes outside, then you
multiply and add. The last number is the remainder, which should be zero.
The numbers represent the coefficients of the quotient. The second to last is the
constant term and the degree increases as you go left.
The quotient is x 2  2 x  15 . Factor this to find the other factors of the polynomial.
Therefore, P( x)  x3  3x 2  13x  15  ( x  1)( x  5)( x  3) .
Note: You can find this by graphing the polynomial. The zeroes are at –1, -5, and 3.
Try. Factor the polynomial x3  8 x 2  19 x  12 by first finding one factor then using
synthetic division.
x3  8 x 2  19 x  12 = ( x  1)( x  3)( x  4) .
If the coefficient of the highest degree term is not 1, then we must use the rational
zero theorem to find a factor of the polynomial.
The factor theorem states that:
If a polynomial P(x) has ax – b as a factor, then P( x)  (ax  b)  Q(b) .
b
b
Substitute x  to show that P    0 .
a
a
b
b
Therefore, if ax – b is a factor of P(x), then P    0 , and if P    0 , ax – b is a
a
a
factor of P(x).
The rational zero theorem states that:
b
is a rational zero of a polynomial P(x) with integral coefficients,
a
then b is a factor of the constant term of the polynomial, and a is a factor
of the coefficient of the highest-degree term in the polynomial.
If x 
Ex. Use the rational zero theorem to find the possible values of
polynomial 3x3  4 x 2  17 x  6 .
Possible values of b:
Possible values of a:
b
for the
a
Possible values of
b
:
a
(Note: exclude any double values, ie. 3/3=1)
It is easier to test integral values first.
Find the other factors using synthetic division or long division.
Factor the quotient.
The factors of the polynomial
Using a graphing calculator to check.
Try. Factor the polynomial 2 x3  3x 2  3x  10 .
2 x3  3x 2  3x  10 = ( x  2)(2 x 2  x  5) Can’t always factor the quotient. How
could you find all the zeroes? Quadratic formula!