1. 3.3 Zeros of polynomial functions
Rational Zero Test: If a polynomial f(x) has
integer coefficients, every rational zero of f has the form
where p and q have no common factors other than 1.
p is a factor of the constant term.
q is a factor of the leading coefficient.
Example: List all possible rational zeros of f(x) = 4x3 + 3x2 – x – 6.
q = 4
p = – 6
Rational Zero Test

2. Graphing Utility: Finding Roots
Find the zeros of
f(x) = 2x3 + x2 – 5x + 2.
f(x) = 2x3 – 9 x2 + 2.
Graphing Utility: Finding Roots

3. Upper and Lower Bounds Theorem
When P(x)÷(x-c) by syn. Div and P(x) with real coefficients.
Upper Bound: a) c>0 and the leading coefficient > 0
x = c is an upper bound for the real zeros if all
numbers in the bottom row are either +ve or zeros
b) c>0 and the leading coefficient < 0
x = c is an upper bound for the real zeros if all
numbers in the bottom row are either -ve or zeros
Lower Bound: c<0
x = c is a lower bound for the real zeros if all
numbers in the bottom row are alternate in sign
( the zero can be considered +ve or –ve)

4. f (x) = – 4 x4 +12 x3+3x2 – 12x + 7. f (x) = 4 x3 – 11x2 – 3x + 15.
Example: According to the Bounds Theorem find both the smallest positive integer that is upper bound and the largest negative integer that is lower bound for the real zeros
f (x) = – 4 x4 +12 x3+3x2 – 12x + 7.
f (x) = 4 x3 – 11x2 – 3x + 15.

5. Descartes’s Rule of Signs
Descartes’s Rule of Signs: If f(x) is a polynomial with real coefficients and a nonzero constant term,
The number of positive real zeros of f is equal to the number of variations in sign of f(x) or less than that number by an even integer.
The number of negative real zeros of f is equal to the number of variations in sign of f(–x) or less than that number by an even integer.
A variation in sign means that two consecutive, nonzero coefficients have opposite signs.
Descartes’s Rule of Signs

6. Example: Descartes’s Rule of Signs
Example: Use Descartes’s Rule of Signs to determine the possible number of positive and negative real zeros of f(x) = 2x4 – 17x3 + 35x2 + 9x – 45.
The polynomial has three variations in sign.
+ to –
+ to –
f(x) = 2x4 – 17x3 + 35x2 + 9x – 45
– to +
f(x) has either three positive real zeros or one positive real zero.
f(– x) = 2(– x)4 – 17(– x)3 + 35(– x)2 + 9(– x) – 45
=2x4 + 17x3 + 35x2 – 9x – 45
One change in sign
f(x) has one negative real zero.
f(x) = 2x4 – 17x3 + 35x2 + 9x – 45 = (x + 1)(2x – 3)(x – 3)(x – 5).
Example: Descartes’s Rule of Signs