PRECALCULUS I

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# PRECALCULUS I - PowerPoint PPT Presentation

PRECALCULUS I. Quadratic Functions. Dr. Claude S. Moore Danville Community College. Polynomial Function. A polynomial function of degree n is where the a ’s are real numbers and the n ’s are nonnegative integers and a n  0 . Quadratic Function.

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### PRECALCULUS I

Functions

Dr. Claude S. MooreDanville Community College

Polynomial Function

A polynomial function of degree n iswhere the a’s are real numbers and the n’s are nonnegative integersand an 0.

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.

Axis of Symmetry

For a quadratic function of the form

gives the axis of symmetry.

Standard Form

A quadratic function of the form

is in standard form.

axis of symmetry: x = hvertex: (h, k)

Characteristics of Parabola

a > 0

vertex: maximum

vertex: minimum

a < 0

### PRECALCULUS I

Higher DegreePolynomial Functions

Dr. Claude S. MooreDanville Community College

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.

graphs of a polynomial function for n odd:

.

an < 0

an > 0

graphs of a polynomial function for n even:

.

an < 0

an > 0

Roots, Zeros, Solutions

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).

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.

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.

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.

### PRECALCULUS I

Polynomial and

Synthetic Division

Dr. Claude S. MooreDanville Community College

Full Division Algorithm

If f (x) and d(x) are polynomialswith d(x)  0 and the degree of d(x) isless than or equal to the degree of f(x),then q(x) and r (x) are uniquepolynomials such thatf (x) = d(x) ·q(x) + r (x)where r (x) = 0 orhas a degree less than d(x).

Short Division Algorithm

f (x) = d(x) ·q(x) + r (x)

dividend quotient divisor remainder

where r (x) = 0 orhas a degree less than d(x).

Synthetic Division

ax3 + bx2 + cx + d divided by x - k

k a b c d

ka

a r

coefficients of quotient remainder

2. Multiply diagonally. 3. Add vertically.

Remainder Theorem

If a polynomial f (x)

is divided by x - k,

the remainder is r = f (k).

Factor Theorem

A polynomial f (x)

has a factor (x - k)

if and only if f (k) = 0.

### PRECALCULUS I

Real Zeros of Polynomial Functions

Dr. Claude S. MooreDanville Community College

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.

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.

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.

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.

Example 3: Rational Zero Test

f(x) = 4x3- 5x2 + 6p  {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- 5x2 + 6 .

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.

Example 4: Upper and Lower Bound

2x3- 3x2- 12x + 8 divided by x + 3

-3 2 -3 -12 8 -6 27 -45

2 -9 15 -37

c = -3 < 0 and #’s in last row alternate positive/negative. Thus, x = -3 is a lower bound to real roots.