Real and Rational Zeros of Polynomials:
(Remember Rational Numbers are numbers that can be expressed in the form
where a and b are integers and b  0 )
a
b
We still have yet to talk about imaginary/complex zeros…..this will be coming up
in future lessons. For now, our only focus is on Real/Rational zeros.
Rational Zero Theorem: If the polynomial P (x)  a n x n  a n 1 x n 1        a1 x  a0
has integer coefficients, then every rational zero of P is of the form:
p
q
Where p is a factor of the constant coefficient a 0
And q is a factor of the leading coefficient a n
Only use the rational zero theorem when a polynomial function is not
factorable!
To find the rational roots of a polynomial:
factors of constent term
1)
Find 
(constant purposely misspelled!)
factors of leading term
2)
Use synthetic division to see if it is truly a zero.
Find all possible rational roots of the following, (do not test to determine if they
are actually zeros):
1)
P( x )  3 x 3  3 x  4
Finding the Rational Zeros of a Polynomial: If polynomial is unfactorable
1)
List Possible Zeros. List all possible rational zeros using the
Rational Zeros Theorem.
2)
Divide. Use synthetic division to evaluate the polynomial at each of
the candidates for the rational zeros that you found in step 1.
When the remainder is 0, note the quotient you have obtained.
3)
Repeat. Repeat steps 1 and 2 for the quotient. Stop when you
reach a quotient that is quadratic or factors easily, and use the
quadratic formula or factor to find the remaining zeros.
(If asked to completely factor the polynomial, use the zeros to rewrite the
polynomial in terms of its factors)
Use the rational zeros theorem to find all real zeros of the following polynomial
function:
2)
P( x)  x 2  2 x  8
3)
P( x)  2 x 3  x 2  13x  6
4)
P( x)  x 4  5 x 3  5 x 2  23x  10
Use the rational theorem to factor each polynomial completely:
5)
P( x)  x 3  6 x  4
6)
P( x)  x 4  5x 3  x 2  21x  18