2.5 Polynomial Zeros (the P/Q Method)
Types of Zeros a Polynomial Can Have
Remember: zeros are found by setting the polynomial = 0 and solving for x.
Real Zeros
... can be either:
Rational
ex) x = 2
x = − 35
x = 4.5
Usually found
by factoring
or
Irrational
ex) x = 4 + 3 and 4 − 3
x = ± 10
Usually found by using
square roots or by using
the quadratic formula
*ALWAYS OCCUR IN
CONJUGATE PAIRS
Complex Zeros
ex) x = 3 + 4 i and 3 − 4 i
x = ± i 33
Usually found by using square roots or quadratic formula
*ALWAYS OCCUR IN CONJUGATE PAIRS
*NEVER SHOW UP AS X‐INTERCEPTS!
For a polynomial you can make a list of its possible RATIONAL
zeros by using its constant term (called P) and its leading
coefficient (called Q).
Possible Rational Zeros =
All ' ± ' factors of P
All ' ± ' factors of Q
ex) List all the possible rational zeros for the polynomial
2 x 4 + 3x 3 − 11x 2 − 9 x + 15
ex) Which of these possible rational zeros show up as actual x‐
intercepts on the graph of f (x) = 2 x 4 + 3x 3 − 11x 2 − 9 x + 15 ?
*Use your calculator to check by using TRACE
ex) Use synthetic division to ‘take out’ each rational zeros and get
the degree of the quotient down to a quadratic to isolate the
remaining zeros.
ex) Find all the zeros for the polynomial
f (x) = 2 x 3 − 3x 2 + 20 x − 30
1st: make the P/Q list
2nd: locate all RATIONAL zeros
by looking at the graph
3rd: use synthetic division
to take out all
rational zeros
4th: set the quotient = 0
and solve for the
remaining zeros.
ex) Find all the zeros for the polynomial
f (x) = x 4 − 4 x 3 − x 2 + 14 x + 10
1st: make the P/Q list
2nd: locate all RATIONAL zeros
by looking at the graph
3rd: use synthetic division
to take out all
rational zeros
4th: set the quotient = 0
and solve for the
remaining zeros.
ex) Construct a polynomial f (x) of degree 4 having these zeros:
x = −4 , x = 35 and x = 2 i
* Make sure it has integer coefficients!
* When finished check to see if f (1) = 50