Useful Fact Sheet Exam 2 –Chapter 2
Definition of a Polynomial Function: is called a polynomial function of degree n. The number , the coefficient of the variable to the highest power, is called the leading coefficient. The maximum number of turning points is given by (n-1)
Leading Term Behavior (Leading Coefficient Test): is the leading term of a polynomial function, then the behavior of the graph as can be described as follows:
If n is even, and If n is even, and If n is odd, and If n is odd, and
Multiplicity of zeros- If r is a zero of even multiplicity (exponent of the zero), then the graph touches the x-axis and turns around at r. If r is a zero of odd multiplicity, then the graph crosses the x-axis at r.
Intermediate Value Theorem – For any polynomial function P(x) with real coefficients, suppose that for a ≠ b, P(a) and P(b) are opposite signs. Then the function has a real zero between a and b.
The Division Algorithm – When we divide a polynomial, P(x), by a divisor, d(x), a polynomial Q(x) is the quotient and a polynomial R(x) is the remainder.
Remainder Theorem – If a number c is substituted for x in the polynomial f(x) then the result f(c) is the remainder that would be obtained by dividing f(x) by x - c. That is , then
The Rational Zero Theorem – If has integer coefficients and (where is reduced to lowest terms) is a rational zero of f, then p is a factor of the constant term and q is a factor of the leading coefficient, .
Properties or Roots of Polynomial Equations
- If a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots.
- If is a root of a polynomial equation with real coefficients (), then the imaginary number is also a root. Imaginary roots, if they exist occur in conjugate pairs.
The Fundamental Theorem of Algebra – If is a polynomial of degree , where , then the equation has at least one complex root (that is at least one zero in the system of complex numbers). Note: that complex numbers include both real numbers and imaginary numbers
The Linear Factorization Theorem – If where and , then where are complex numbers.
In words: An nth degree polynomial can be expressed as the product of a nonzero constant and n linear factors, where each linear factor has a leading coefficient of 1
Determining a Vertical Asymptotes – For a rational function
, if is a zero of the denominator, then the line is a vertical asymptote.
Determining a Horizontal Asymptotes – For a rational function , the degree of the numerator is n and the degree of the denominator is m.
- if , the x-axis, or , is the horizontal asymptote of the graph of f.
- if , the line, or , is the horizontal asymptote of the graph of f.
- if , the graph has no horizontal asymptote.
Slant Asymptotes – In general if , p and q have no common factors, and the degree of p is one degree higher than the degree of q, we find a slant (or oblique) asymptote by dividing q(x) into p(x). The division will take the form The equation of the slant asymptote is obtained by dropping the term with the remainder. Thus the equation of the slant asymptote is
Descartes’s Rule of Signs – Let P(x), written in decreasing exponential order or ascending exponential order
The number of positive real zeros of P(x) is either:
- the same as the number of variations of sign in P(x)
- Less than the number of sign variations of sign in P(x), by a positive even integer. If P(x) has only one variation in sign then P has exactly one positive real zero.
The number of negative real zeros of P(x) is either:
- the same as the number of variations of sign in P(-x)
- Less than the number of sign variations of sign in P(-x), by a positive even integer. If P(x) has only one variation in sign then P has exactly one negaitve real zero.
Modeling using Variation:
English Statement / Equationy varies directly as x
y is directly proportional to x /
y varies directly as
y is directly proportional to /
y varies inversely as
y is inversely proportional to /
y varies inversely as
y is inversely proportional to /
Y varies jointly as x and z /
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