MTH 165 COLLEGE ALGEBRA

Creating open expressions and using those expressions to write equations involving one or two variables to solve problems and applications will be integrated throughout this course (especially in * topics). Examples from other disciplines will be incorporated whenever possible. A comprehensive departmental final exam testing the degree of mastery of the following course objectives is required.

 1. Sets of Numbers

1.1  Review classifying a given real number as being a counting or natural number, whole number, integer, rational or irrational number.

1.2  Review the concept of complex numbers.

1.3  Review writing complex numbers in  a+bi  form; review adding, subtracting, multiplying, and dividing complex numbers in  a+bi  form.

*2. Equation Solving Techniques

2.1  Review solving linear equations and formulas for a single variable.

2.2  Solve quadratic equations in one variable.

a. Review solving by factoring.
b. Review solving by the Square Root Method.
c. Review solving by completing the square on ax² + bx +c = 0, where a = 1.
d. Solve by completing the square on ax² + bx +c = 0,
where a ≠ 1.
e. Review solving by the quadratic formula.
f.  Review using the discriminant to classify the roots.

2.3  Review solving rational equations.

2.4  Solve radical equations.

a.  Review solving radical equations with one radical term.
b.  Solve radical equations with at most two radical terms.

2.5  Solve problems involving direct, joint, and/or inverse variation.

2.6  Solve equations involving rational exponents.

2.7  Solve equations that are quadratic in form.

 3. Polynomials

3.1  Review multiplication and division of polynomials.

3.2  Perform synthetic division using rational divisors.

3.3  Use synthetic division to find the quotient and remainder of the polynomial function P(x) when P(x) is divided by x - c, to determine whether a given number is a zero of P(x), and to determine whether x - c is a factor of
P(x).

3.4  Use the Remainder Theorem to find remainder P(c) when a polynomial function P(x) is divided by x - c.

3.5  Use the Fundamental Theorem of Algebra to find a polynomial function  P(x) with n given zeros.

3.6  Given a polynomial function and some of its zeros, find the remaining zeros.

3.7  Use the Rational Zero Theorem to find the rational zeros of a polynomial function with integer coefficients.

 4. Factoring

4.1  Review factoring techniques, including GCF factoring, factoring by grouping, factoring the difference of two squares, factoring trinomials of the form ax² + bx +c, factoring perfect square trinominals, factoring the sum and difference of two cubes.

 5. Rational Expressions

5.1  Review arithmetic operations on rational expressions.

5.2  Review simplification of complex fractions.

 6. Inequalities in One Variable

6.1  Review solving compound linear inequalities, expressing the solution using set-builder notation, interval notation, and on the real number line

6.2  Solve factorable quadratic inequalities, and inequalities involving rational expressions, expressing the solution using set-builder notation, interval notation, and on the real number line.

*7. Absolute Value

7.1 State the definition of absolute value.

7.2 Review solving a linear absolute value equation in one variable.

7.3 Review solving a linear absolute value inequality in one variable, expressing the solution using set-builder notation, interval notation and on the real number line.

*8. Functions

8.1  Determine whether a relation is a function.

8.2  Find the domain of a given function algebraically.

8.3  Find the domain and range of a function from its graph.

8.4  Determine algebraically and graphically whether a given relation is a function.

8.5  Review the use of function notation.

8.6  Perform combinations of functions (f + g, f - g, f · g, f / g).

8.7  Find the composition of functions (f o g).

8.8  Determine whether a given function is one-to-one.

8.9  Given a one-to-one function f(x), find the inverse function, f ^-1(x), algebraically and graphically.

8.10  Determine the zeros of a function.

*9. The Cartesian Coordinate System

9.1  Review writing the equation of a line using the slope-intercept form, and the point-slope form.

9.2  Use the vertex, axis of symmetry, and intercepts to graph a quadratic function of the form f(x) = ax² + bx +c (REVIEW) or f(x) = a(x - h)² + k (NEW)

9.3  Given the equation of a circle, determine the center and radius, and use these to sketch the graph of the circle.

9.4  Determine algebraically and graphically whether the graph of a given equation is symmetric with respect to the x-axis, y-axis, and/or origin.

9.5  Identify the graphs of the following types of functions: linear, quadratic, cubic, absolute value, square root, exponential, and logarithmic.

9.6  Use translations to sketch a graph which is a horizontal and/or vertical shift from a graph of the following types of functions: linear, quadratic, cubic, absolute value, square root, exponential, and logarithmic.

9.7  Graph the following non-linear equations in two variables using intercepts and by obtaining a finite number of ordered pairs:

• Absolute value functions
• Equations of the type x = ay² + by + c

9.8  Graph piece-wise functions with constant, linear, and/or quadratic pieces.

9.9  Review graphing a linear inequality in two variables.

9.10  Graph a quadratic inequality of the form y > ax² + bx + c. (also <, ≥, ≤ )

9.11  Sketch the graphs of polynomial functions of no more than fourth degree.

*10. Systems of Equations and Inequalities

10.1  Solve linear-quadratic systems of equations in two variables algebraically.

10.2  Solve linear-quadratic systems of equations in two variables graphically.

10.3  Solve systems of two quadratic equations in two variables graphically.

10.4  Solve systems of linear and/or quadratic inequalities in two variables graphically.

11. Exponents and Radicals

11.1  Review simplifying exponential expressions with rational exponents.

11.2  Review simplifying radical expressions.

11.3  Review performing arithmetic operations on radical expressions.

*12. Exponential and Logarithmic Functions

12.1  Define exponential functions of the form f(x) = a (b^cx), b > 0,
b ≠ 1, c ≠ 0.

12.2  Graph equations of the form f(x) = a (b^cx), b > 0, b ≠ 1, c ≠ 0.

12.3  Graph equations of the form f(x) = e^cx and f(x) = 10^cx, c ≠ 0.

12.4  Define logarithmic functions of the form f(x) = a log_b x, b > 0, b ≠ 1.

12.5  Graph equations of the form f(x) = a log_b x, b > 0, b ≠ 1.

12.6  Graph equations of the form f(x) = ln x and f(x) = log x.

12.7  Express logarithmic equations in exponential form.

12.8  Express exponential equations in logarithmic form.

12.9  Use the properties of logarithms to rewrite expressions involving logarithms.

12.10  Use a calculator to evaluate common and natural logarithms and antilogarithms.

12.11  Solve exponential equations and logarithmic equations.

12.12  Solve applied problems involving exponential and logarithmic functions.

(revised 03/26/04)