College Algebra Midterm Study Guide: Key Concepts and Problem Types

Algebraic Expressions

Evaluating Expressions

Algebraic expressions can be evaluated by substituting given values for variables and following the order of operations.

• Key Point: Substitute the values for each variable and simplify step by step.

• Example: For $|p| + 18/(|1|/|q|)$ where $p = -6$ and $q = 1$, substitute and simplify.

Exponents and Percentages

Converting Decimals to Percentages

To convert a decimal to a percentage, multiply by 100 and add the percent symbol (%).

• Key Point: $0.9$ as a percentage is $0.9 imes 100 = 90\%$.

• Example: $0.75$ as a percentage is $75\%$.

Polynomials

Writing and Simplifying Algebraic Expressions

Algebraic phrases can be translated into expressions using variables and operations.

• Key Point: "The number $y$ decreased by the sum of eighteen and this number" translates to $y - (18 + y)$.

• Example: "Twice a number decreased by five" is $2x - 5$.

Classifying Polynomials

Polynomials are classified by degree and number of terms.

• Degree: The highest exponent of the variable.

• Terms: Monomial (1 term), Binomial (2 terms), Trinomial (3 terms).

• Example: $5y^4 + 8y^2 - 12$ is a trinomial of degree 4.

Factoring Polynomials

Factoring involves expressing a polynomial as a product of its factors, often by taking out the greatest common factor (GCF).

• Key Point: Factor out the lowest power of each variable.

• Example: $11d^7 - 33d^9 = 11d^7(1 - 3d^2)$.

Radical Expressions

Simplifying Radical Expressions

Radical expressions can be simplified by factoring under the radical and reducing.

• Key Point: $rac{8}{ ext{11} imes2 ext{sqrt}(6)}$ can be rationalized by multiplying numerator and denominator by $ext{sqrt}(6)$.

• Example: $rac{1}{ ext{sqrt}(2)} = rac{ ext{sqrt}(2)}{2}$.

Rational Exponents

Rational exponents represent roots and powers. Negative exponents indicate reciprocals.

• Key Point: $a^{5}/a^{7} = a^{-2} = 1/a^{2}$.

• Example: $x^{-3} = 1/x^{3}$.

Linear Equations

Solving Word Problems with Linear Equations

Linear equations can be used to solve real-world problems involving investments, rates, and totals.

• Key Point: Set up equations based on the problem statement and solve for unknowns.

• Example: If $x$ is invested at 5% and $y$ at 3%, and total interest is $17,000$, set up $0.05x + 0.03y = 17,000$.

Rational Equations

Inverse Variation

Inverse variation occurs when one variable increases as the other decreases, such that their product is constant.

• Key Point: $v imes d = k$, where $v$ is speed, $d$ is diameter, and $k$ is constant.

• Example: If $v_1 d_1 = v_2 d_2$, solve for unknowns.

Imaginary and Complex Numbers

The Imaginary Unit

The imaginary unit $i$ is defined as $i = ext{sqrt}(-1)$. Powers of $i$ cycle every four terms.

• Key Point: $i^2 = -1$, $i^3 = -i$, $i^4 = 1$.

• Example: $i^{12} = (i^4)^3 = 1$.

Complex Numbers

Complex numbers are of the form $a + bi$, where $a$ and $b$ are real numbers.

• Key Point: Division by $i$ can be simplified using $i^2 = -1$.

• Example: $119 / i = 119i / i^2 = 119i / -1 = -119i$.

Quadratic Equations

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: For $2x^2 - x + 1 = 0$, $a = 2$, $b = -1$, $c = 1$.

Square Root Property

For equations of the form $x^2 = a$, solutions are $x = \pm \sqrt{a}$.

• Example: $x^2 = 9$ has solutions $x = 3$ and $x = -3$.

Completing the Square

Completing the square is a method to solve quadratic equations by rewriting them in the form $(x + p)^2 = q$.

• Key Point: Add and subtract the same value to create a perfect square trinomial.

• Example: $x^2 + 6x + 9 = 0$ can be written as $(x + 3)^2 = 0$.

Geometry in Algebra

Applying Algebra to Triangles

Algebraic equations can represent geometric properties, such as the Pythagorean theorem for right triangles.

• Key Point: For a right triangle with sides $a$, $b$, and hypotenuse $c$, $a^2 + b^2 = c^2$.

• Example: If sides are $x$, $x + 9$, and $3x - 3$, set up $x^2 + (x + 9)^2 = (3x - 3)^2$.

Summary Table: Polynomial Classification

Expression	Degree	Number of Terms	Type
$5y^4 + 8y^2 - 12$	4	3	Trinomial
$x^2 - 4$	2	2	Binomial
$7x$	1	1	Monomial

Additional info:

• Some problems involve translating word problems into algebraic equations, a key skill in College Algebra.

• Imaginary and complex numbers are introduced, including operations and simplification.

• Quadratic equations are solved using multiple methods, including the quadratic formula and completing the square.