CHAPTER 2 TOPICS

Polynomial Function

Let n be a non negative integer and let   be real numbers with
The function given by: is called
polynomial function of x with degree n. (page 128)

Quadratic Function
Let, a , b , c be real numbers with a ≠ 0 . The function given by f (x) = ax^2 + bx + c is
called a quadratic function. (page 128)

Standard form of a Quadratic Function
The quadratic function given by f (x) = a(x − h)^2 + k, a ≠ 0is in standard form. The graph
of f is a parabola whose axis is the vertical line x = h and whose vertex is the point
(h, k) if a > 0 , the parabola opens upward, and if a < 0 , the parabola opens downward.
(page 131)

Vertex of a Parabola
The vertex of a graph of

1. If a < 0 , has a minimum at

2. If a > 0 , has a maximum at

(page 133)

The Remainder Theorem
If a polynomial f(x) is divided by x – k, the remainder is r = f(k). (page 157)

The Factor Theorem
A polynomial f(x) has a factor (x-k) if and only if f(x)=0. (page 157)

Definition of a Complex Number
If a and b are real numbers, the number a+bi is a complex number, and it is said to be
written in the standard form. If b=0, the number a+bi =a is a real number. if b ≠ 0 , the
number a+bi is called an imaginary number. A number of the form bi, were b ≠ 0 , is
called pure imaginary number. (page 162)

Complex Conjugates
A pair of complex numbers of the form a+bi and a-bi, are called complex conjugates.
(page 165)

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 . (page 169)

Linear Factorization Theorem
If f(x) is a polynomial of degree n, where n>0, then f has precisely n linear factors
  where are complex numbers. (page 169)

The Rational Zero Test
If the polynomial has integer coefficients,
every rational zero of f has the form

where p and q have no common factors other than 1, and
p = a factor of the constant term a ₀
q = a factor of the leading coefficient a _n. (page 170)

Complex Zero Occur in Conjugate Pairs
Let f(x) be a polynomial function that has real coefficients. If a+bi, where b ≠ 0 , is a zero
of the function, the conjugate a-bi is also a zero of the function. (page 173)

Viète Relations
If is a polynomial of degree n ≥1 with complex coefficients (so the numbers
are complex with ), by the fundamental theorem of algebra P (x)
has n (not necessarily distinct) complex roots . Viète's formulas state that

In other words, the sum of all possible products of k roots of P(x) (with the indices in
each product in increasing order so that there are no repetitions) equals
for each

for each k = 1, 2, … , n

For a polynomial P(x) with degree n = 2

Viète formulas state that the solutions x₁ and x₂ Of the equation P(x)=0 satisfy

Vertical Asymptotes
The line x = a is a vertical asymptote of the graph of f if

where x→a , either from the right or from the left. (page 185)

Horizontal Asymptote
The line y = b is a horizontal asymptote of the graph f if

f (x)→b

as x→∞ or x→−∞. (page 185)

Slant Asymptotes
If the degree of numerator is exactly one more than the degree of the denominator the
graph of the function has a slant (or oblique) asymptote. (page 190)