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Polynomial Function

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

Standard Form

A quadratic function of the form

is in standard form.

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

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.

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.

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

1. Copy leading coefficient.

2. Multiply diagonally. 3. Add vertically.

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.

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