CLEP College Algebra Flashcards

Algebraic Concepts

• Additive Inverse: The additive inverse of a number 'a' is -a.
• Multiplicative Inverse: The multiplicative inverse of a non-zero number 'a' is 1/a.
• Negative Exponent Rule: Raising an expression 'b' to a negative exponent 'n' is equivalent to 1/b^n.
• Exponent Addition: When multiplying expressions with the same base, x^n * x^m = x^(m+n).
• Exponent Raising: (x^m)^n simplifies to x^(m*n).
• Exponent Subtraction: For division, x^m / x^n = x^(m-n).

Advanced Exponents and Radicals

• Fractional Exponents: x^(1/2) represents the square root of x, and x^(m/n) denotes the n-th root of x^m.
• Multiplying Radicals: The product of two square roots is √a * √b = √(ab).

Factoring Techniques

• Difference of Squares: x^2 - y^2 factors into (x - y)(x + y).
• Perfect Squares: x^2 + 2xy + y^2 = (x + y)^2 and x^2 - 2xy + y^2 = (x - y)^2.
• Cubes Factoring: x^3 + y^3 = (x + y)(x^2 - xy + y^2) and x^3 - y^3 = (x - y)(x^2 + xy + y^2).

Slope and Line Equations

• Slope from Points: Given two points, the slope (m) is calculated as m = (y2 - y1) / (x2 - x1).
• Slope-Intercept Form: The equation of a line can be expressed as y = mx + b, where m is the slope and b is the y-intercept.
• Point-Slope Form: For a line through point (x1, y1) with slope m: (y - y1) = m(x - x1).
• Intercept Form: Expressed as x/a + y/b = 1, where a and b are x and y intercepts respectively.

Graphing and Properties

• Perpendicular Line Slopes: Slopes of perpendicular lines are negative reciprocals of each other.
• Discriminant: Used to determine the number of real solutions:
  • If b^2 - 4ac > 0, there are 2 real solutions.
  • If b^2 - 4ac = 0, there is 1 real solution.
  • If b^2 - 4ac < 0, there are no real solutions but 2 imaginary solutions.

Logarithmic Operations

• Addition of Logarithms: log a + log b = log(ab).
• Subtraction of Logarithms: log a - log b = log(a/b).
• Power Rule: log a^b = b log a.

Sequences

• Arithmetic Sequence: Defined as an_n = a1 + (n - 1)d, where d is the common difference.
• Sum of Arithmetic Sequences: Calculated using Sn = n/2[2a1 + (n - 1)d] or Sn = n/2(a1 + a2).
• Geometric Sequence: An expression a_n = a1 * r^(n-1), where r is the common ratio.
• Sum of Geometric Sequences: Calculated as Sn = [a1(1 - r^n)] / (1 - r).
• Infinite Geometric Series: Sn = a1 / (1 - r), valid for |r| < 1.

Combinatorics and Linear Algebra

• Factorial: Defined as n! = n(n - 1)(n - 2)...321.
• Combination Formula: nCr = n! / (r!(n - r)!).
• Determinant of a 2x2 Matrix: For a matrix with elements a, b, c, d, the determinant is ad - bc.

Additional Concepts

• Vertical Line Test: Used to determine whether a graph represents a function.
• Domain Determination: Domain exceptions occur when:
  • The denominator equals zero.
  • An even root yields a negative number.
  • The logarithm of zero or a negative number occurs.
• Graph Symmetry: Functions are even if symmetrical about the y-axis and odd if symmetrical about the origin.
• Transformations: Adjustments to functions include vertical shifts, horizontal shifts, and reflections across axes.

Polynomial Roots

• Root Analysis: The number of roots corresponds to how many times the graph touches or crosses the x-axis.
• Max and Min Roots:
  • For a polynomial of degree n, the maximum number of real roots is n.
  • If n is even, the minimum is 0; if n is odd, the minimum is 1.