1. Dr. Claude S. Moore Danville Community College
PRECALCULUS I
Quadratic
Functions
Dr. Claude S. Moore Danville Community College

2. Polynomial Function
A polynomial function of degree n is where the a’s are real numbers and the n’s are nonnegative integers and an  0.

3. a, b, and c are real numbers and a  0.
Quadratic Function
A polynomial function of degree 2 is called a quadratic function.
It is of the form
a, b, and c are real numbers and a  0.

4. Axis of Symmetry For a quadratic function of the form
gives the axis of symmetry.

5. axis of symmetry: x = h vertex: (h, k)
Standard Form
A quadratic function of the form
is in standard form.
axis of symmetry: x = h vertex: (h, k)

6. Characteristics of Parabola
vertex: maximum
vertex: minimum
a < 0

7. PRECALCULUS I Higher Degree Polynomial Functions
Dr. Claude S. Moore Danville Community College

8. Characteristics The graph of a polynomial function… 1. Is continuous.
2. Has smooth, rounded turns.
3. For n even, both sides go same way.
4. For n odd, sides go opposite way.
5. For a > 0, right side goes up.
6. For a < 0, right side goes down.

9. Leading Coefficient Test: n odd
graphs of a polynomial function for n odd: .
an > 0
an < 0

10. Leading Coefficient Test: n even
graphs of a polynomial function for n even: .
an > 0
an < 0

11. Roots, Zeros, Solutions 1. x = a is root or zero of f.
The following statements are equivalent for
real number a and polynomial function f :
1. x = a is root or zero of f.
2. x = a is solution of f (x) = 0.
3. (x - a) is factor of f (x).
4. (a, 0) is x-intercept of graph of f (x).

12. Repeated Roots (Zeros)
1. If a polynomial function contains a factor (x - a)k, then x = a is a repeated root of multiplicity k.
2. If k is even, the graph touches (not crosses) the x-axis at x = a.
3. If k is odd, the graph crosses the x-axis at x = a.

13. Intermediate Value Theorem
If a < b are two real numbers
and f (x)is a polynomial function
with f (a)  f (b),
then f (x) takes on every real
number value between
f (a) and f (b) for a  x  b.

14. NOTE to Intermediate Value
Let f (x) be a polynomial function and a < b be two real numbers.
If f (a) and f (b)
have opposite signs
(one positive and one negative),
then f (x) = 0 for a < x < b.

15. Dr. Claude S. Moore Danville Community College
PRECALCULUS I
Polynomial and
Synthetic Division
Dr. Claude S. Moore Danville Community College

16. Full Division Algorithm
If f (x) and d(x) are polynomials with d(x)  0 and the degree of d(x) is less than or equal to the degree of f(x), then q(x) and r (x) are unique polynomials such that f (x) = d(x) ·q(x) + r (x) where r (x) = 0 or has a degree less than d(x).

17. Short Division Algorithm
f (x) = d(x) ·q(x) + r (x)
dividend quotient divisor remainder
where r (x) = 0 or has a degree less than d(x).

18. ax3 + bx2 + cx + d divided by x - k
Synthetic Division
ax3 + bx2 + cx + d divided by x - k
k a b c d ka
a r
coefficients of quotient remainder
1. Copy leading coefficient.
2. Multiply diagonally. 3. Add vertically.

19. the remainder is r = f (k).
Remainder Theorem
If a polynomial f (x)
is divided by x - k,
the remainder is r = f (k).

20. Factor Theorem A polynomial f (x) has a factor (x - k)
if and only if f (k) = 0.

21. PRECALCULUS I Real Zeros of Polynomial Functions
Dr. Claude S. Moore Danville Community College

22. Descartes’s Rule of Signs
a’s are real numbers, an  0, and a0  0.
1. Number of positive real zeros of f equals number of variations in sign of f(x), or less than that number by an even integer.
2. Number of negative real zeros of f equals number of variations in sign of f(-x), or less than that number by an even integer.

23. Example 1: Descartes’s Rule of Signs
a’s are real numbers, an  0, and a0  0.
1. f(x) has two change-of-signs; thus, f(x) has two or zero positive real roots.
2. f(-x) = -4x3 - 5x2 + 6 has one change-of-signs; thus, f(x) has one negative real root.

24. Example 2: Descartes’s Rule of Signs
Factor out x; f(x) = x(4x2 - 5x + 6) = xg(x)
1. g(x) has two change-of-signs; thus, g(x) has two or zero positive real roots.
2. g(-x) = 4x2 + 5x + 6 has zero change-of-signs; thus, g(x) has no negative real root.

25. Rational Zero Test
If a’s are integers, every rational zero of f has the form
rational zero = p/q,
in reduced form, and p and q are factors of a0 and an, respectively.

26. Example 3: Rational Zero Test
f(x) = 4x3 - 5x p  {1, 2, 3, 6}
q  {1, 2, 4}
p/q  {1, 2, 3, 6, 1/2, 1/4, 3/2, 3/4} represents all possible rational roots of f(x) = 4x3 - 5x

27. Upper and Lower Bound
f(x) is a polynomial with real coefficients and an > 0 with f(x)  (x - c), using synthetic division:
1. If c > 0 and each # in last row is either positive or zero, c is an upper bound.
2. If c < 0 and the #’s in the last row alternate positive and negative, c is an lower bound.

28. Example 4: Upper and Lower Bound
2x3 - 3x2 - 12x + 8 divided by x + 3
c = -3 < 0 and #’s in last row alternate positive/negative. Thus, x = -3 is a lower bound to real roots.