Day 5: The Factor Theorem
The ____________ ________________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 .
Ex: State whether 2x + 3 is a factor of 6 x3  9 x 2  2 x  3 .
Ex:
a) What number if inputed into the polynomial x 3  8 x 2  19 x  12 would result in zero?
b) Then what is a factor of the above polynomial (a divisor that will result in a remainder of zero)
Integral Zero Theorem
The ______________ _________ _____________ 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: Factor the polynomial x3  3x 2  13x  15 by performing the following steps.
a) Find the possible values of b (therefore the possible factors) for the polynomial.
b) Calculate the value of the polynomial using the different options until you find one that results in
zero.
c) Determine a factor of the polynomial.
Once we have found one factor we can use ____________ _______________ (or _________ division)
to factor the polynomial further.
Performing long division for above example
Performing synthetic division for above example
The coefficients of the polynomial go on top, the value of b goes outside, then after you drop the first
coefficient, you multiply and add. The results will be the coefficients of the quotient statement with the
last number being the remainder (the remainder should be zero since we should have a factor). The
second to last term is the constant term and the degree increases as you go left.
The fully factored form of the above example would be
Ex: Factor the polynomial x3  8 x 2  19 x  12 by first finding one factor then using synthetic division.
Note: You can find the factors of a
polynomial by graphing it. The zeroes
are at –1, –3, and –4.
Recall that when a polynomial P(x) is divided by ax – b, and the remainder is a constant, then the
b
b
remainder is P   . If P   0 , it must be that ax – b is a factor of the polynomial according to the
a
a
Factor Theorem.
If the coefficient of the highest degree term is not 1, then we must use the __________ ________
_____________ to find a factor of the polynomial.
The rational zero theorem states that:
b
is a rational zero of a polynomial P(x) with integral coefficients, then b is a factor
a
of the constant term of the polynomial, and a is a factor of the coefficient of the highestdegree term in the polynomial.
b
Ex: Use the rational zero theorem to find the possible values of
for the
a
polynomial 3x3  4 x 2  17 x  6 . The following steps must be done in order to use the rational zero
theorem:
If x 
i) Determine possible values of b:
iii) Determine possible values of
vi) Input different values of
ii) Determine possible values of a:
b
:
a
(Note: exclude any double values, ie. 3/3=1)
b
for x in order to determine a factor.
a
v) Find the other factors using synthetic division or long division.
vi) Factor the quotient if possible.
vii) Write the factored form of the polynomial.
viii) Use a graphing calculator to check answers
Ex: Factor the polynomial 2 x3  3x 2  3x  10 .
Note: You may not be able to factor a polynomial completely.
Summary of the theorems
Polynomials that have integral coefficients can have the form of
P( x)  a n x n  a n 1 x n 1 ...  a1 x1  a 0
where an = 1
Integral Zero Theorem
If x  b is a factor of the
polynomial, then b must be a factor
of the constant term.
or
P( x)  a n x n  a n 1 x n 1 ...  a1 x1  a 0
where an  1
Rational Zero Theorem
If ax  b is a factor of the polynomial,
then b must be a factor of the
constant term and a must be a factor
of the leading coefficient.
Factor Theorem
If P(b)  0 , then x – b is a factor.
b
If P ( )  0 , then ax – b is a factor.
a
Note: The above three theorems help factor polynomials. The Remainder Theorem simply helps find
remainders when dividing polynomials.
Remainder Theorem
The remainder when dividing P(x)
by x – b is P(b).
The remainder when dividing P(x) by
b
ax – b is P   .
a