Polynomial and Rational Functions

1. Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial f(x) by (x - a) is equal to f(a).

• Key Point: To find the remainder when dividing f(x) by (x - a), simply evaluate f(a).

• Example: Find the remainder when f(x) = 5x^4 + 8x^3 + 2x^2 + 4 is divided by (x - 1):

  • Compute f(1) = 5(1)^4 + 8(1)^3 + 2(1)^2 + 4 = 5 + 8 + 2 + 4 = 19.

  • The remainder is 19.

2. Factor Theorem

The Factor Theorem is a special case of the Remainder Theorem. It states that (x - a) is a factor of f(x) if and only if f(a) = 0.

• Key Point: If f(a) = 0, then (x - a) divides f(x) exactly.

• Example: Does (x - 3) divide f(x) = x^3 - 8x^2 - 11x^2 + 4x + 6?

  • Compute f(3). If f(3) = 0, then (x - 3) is a factor.

3. Descartes' Rule of Signs

Descartes' Rule of Signs is used to determine the possible number of positive and negative real zeros of a polynomial function.

• Key Point: The number of positive real zeros is equal to the number of sign changes in f(x) or less than that by an even number.

• The number of negative real zeros is determined by the sign changes in f(-x).

• Example: For f(x) = 5x^5 + 3x^4 + 2x^3 + x^2 - 2x - 5, count the sign changes in the coefficients to estimate the number of positive real zeros.

4. Turning Points of a Polynomial

The maximum number of turning points of a polynomial function of degree n is n - 1.

• Key Point: A turning point is where the graph changes direction from increasing to decreasing or vice versa.

• Example: For f(x) = x^5 + x^2 - 2x^4 + 5, the degree is 5, so the maximum number of turning points is 4.

Rational Functions

5. Domain of a Rational Function

The domain of a rational function is all real numbers except where the denominator is zero.

• Key Point: Set the denominator equal to zero and solve for excluded values.

• Example: For k(x) = (x - 2)/(x^2 + x - 2), set x^2 + x - 2 = 0 and solve for x.

6. Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches infinity or negative infinity.

• Key Point: Compare the degrees of the numerator and denominator:

  • If degree numerator < degree denominator: asymptote at y = 0.

  • If degrees are equal: asymptote at y = (leading coefficient numerator)/(leading coefficient denominator).

  • If degree numerator > degree denominator: no horizontal asymptote.

• Example: For k(x) = (9x)/(x - 1), both degrees are 1, so the asymptote is y = 9/1 = 9.

7. Vertical Asymptotes

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at those points).

• Key Point: Set the denominator equal to zero and solve for x.

• Example: For k(x) = (9x)/(x - 1), the vertical asymptote is at x = 1.

8. Intercepts of Rational Functions

Intercepts are points where the graph crosses the axes.

• y-intercept: Set x = 0 and solve for k(0).

• x-intercept: Set the numerator equal to zero and solve for x.

• Example: For k(x) = (9x)/(x - 1):

  • y-intercept: k(0) = 0

  • x-intercept: 9x = 0 so x = 0

9. Graphing Rational Functions

To graph a rational function, identify its intercepts, asymptotes, and domain, then plot key points and sketch the curve.

• Key Steps:

  1. Find the domain.

  2. Find intercepts.

  3. Find vertical and horizontal asymptotes.

  4. Plot additional points as needed.

  5. Sketch the graph, showing behavior near asymptotes.

• Example: For k(x) = (9x)/(x - 1), plot the vertical asymptote at x = 1, horizontal asymptote at y = 9, and intercept at (0, 0).

10. Using Graphs to Determine Domain and Range

The domain is all x-values for which the function is defined; the range is all possible y-values the function can take.

• Key Point: Use the graph to identify excluded x-values (vertical asymptotes) and y-values that the function cannot reach (horizontal asymptotes, if any).

• Example: For a graph with a vertical asymptote at x = 1 and a horizontal asymptote at y = 9, the domain is all real numbers except x = 1, and the range is all real numbers except y = 9.

Summary Table: Asymptotes and Intercepts of Rational Functions

Additional info: The above notes expand on the brief questions and provide context, definitions, and examples for each concept relevant to College Algebra.