Section 2.1 Vocabulary

Definition of Polynomial function • Let n be a nonnegative integer and let a n ,a n-1 , …, a 2 , a 1 , a 0 be real numbers with a n not equal to zero. The function given by F(x) = a n x n + a n-1 x n-1 +….. + a 2 x 2 + a 1 x + a 0 Is called a polynomial function in x of degree n.

Definition of Quadratic • Function Let a, b, and c be real numbers with a not equal to zero. The function given by F(x) = ax 2 + bx + c is called a quadratic function

The graph of a quadratic function is a special type of U-shaped curve called a parabola.

All parabolas are symmetric with respect to a line called the axis of symmetry.

Standard form of a Quadratic Function • The quadratic function given by F(x) = a(x – h) 2 + k , where a cannot equal zero

Maximum and Minimum values of Quadratic Functions • If a > 0, f has a minimum value at x = -b / (2a) • If a < 0 , f has a maximum value at x = -b/(2a)

Section 2.2 Vocabulary

Continuous

• In order to be continuous the graph of the polynomial function has no breaks, holes, or gaps.

Leading Coefficient Test • If the highest exponent is odd, and the leading coefficient is positive: • Falls left, rises right If the highest exponent is odd, and the leading coefficient is negative: • Rises left falls right If the highest exponent is even, and the leading coefficient is positive: • Rises left and right If the highest exponent is even, and the leading coefficient is negative: Falls left and right

A polynomial a n x n + …+ a 1 x + a 0 has at most n real zeros, and at most n-1 relative extrema( minima, or maxima)

Real Zeros of Polynomial Functions • • • • If a is a real number than the following statements are equivalent: x= a is a real zero of the function X = a is a solution of the polynomial equation f(x) = 0 (x-a) is a factor of the polynomial f(x). (a,0) is an x-intercept of the graph of f(x).

Repeated zeros

• For a polynomial function, a factor of (x – a) k , k > 1, yields a repeated zero x = a of multiplicity k. 1. If k is odd, the graph crosses the x-axis at x = a. 1. If k is even, the graph touches the x-axis at x = a

•

Intermediate Value Theorem

Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a) ≠ f(b), then in the interval [a,b], f takes on every value between f(a) and f(b).

Section 2.3 Vocabulary

2 methods of dividing polynomials

• Long Division • Synthetic Division(only when the divisor has the form x – k)

•

Division algorithm

If f(x) and d(x) are polynimials where d(x) ≠ 0, and the degree of d(x) is less than or equal to the degree of f(x), there exists unique polynomials q(x) and r(x) such that: f(x) = d(x) q(x) + r(x) Note: f(x) / d(x) is improper r(x) / d(x) is proper

The remainder theorem

• If a polynomial f(x) is divided by x – k, the remainder is r = f(k).

The Factor Theorem

•

A polynomial f(x) has a factor of (x – k) if and only if f(k) = 0

Descartes' Rule of Signs

• Let f(x) = a n x n + …+ a 1 x + a 0 be a polynomial with real coefficients and a 0 ≠ 0. 1. The number of positive real zeros of f is either equal to the number of variations of the sign of f(x) or less than that number by an even integer 2. The number of negative real zeros of f is wither equal to the number of variations of the sign of f(-x) or less than that number by an even integer

A variation in sign means that two consecutive (non zero) coefficients have opposite signs.

A real number b is an upper bound for the real zeros of f is no zeros are greater than b. Similarly, b is a lower bound if no real zeros of f are less than b.

Section 2.4 Vocabulary

Complex number

Has a real part(a) and an imaginary part (bi) and is written in

standard form:

a + bi

Equality of complex numbers

• Two complex numbers a + bi and c + di, written in standard form, are equal to each other a + bi = c + di If and only if a = c and b = d

Pairs of complex numbers of the forms a + bi and a – bi are called complex

conjugates

Section 2.5 Vocabulary

The Fundamental Theorem of Algebra

If f(x) is a polynomial of degree n, where n > 0, then f has at least one zero in the complex number system.

Linear Factorization Theorem

If f(x) is a polynomial of degree n, there n > 0, f has precisely n linear factors f(x) = a n (x – c 1 ) (x – c 2 ) …..( x – c n ) Where c 1 , c 2 ,…., c n numbers. are complex

A quadratic factor with no real zeros is said to be prime or irreducible over the reals.

Section 2.6 Vocabulary

A rational function can be written in the form: f(x) = N(x) / D(x)

The line x = a is a vertical asymptote of the graph of f if f(x) approaches infinity or f(x) approaches negative infinity as x approaches a, either from the right or from the left.

The line y = b is a horizontal asymptote of the graph of f if f(x) approaches b as x approaches positive or negative infinity.

Section 2.7 Vocabulary

Consider a rational function. If the degree of the numerator is exactly one more than the degree of the denominator then the function has a slant

asymptote.