Math

Measures of Central Tendency

Measures of central tendency describe the center of a data set using a single representative value.

• Mean

  • The arithmetic average of a data set.

  • Calculated by summing all values and dividing by the total number of values $n$.

  • Formula: $\bar{x} = \frac{\sum x}{n}$

• Median

  • The middle value when data points are arranged in ascending or descending order.

  • If the number of data points $n$ is odd, the median is the value at position $\frac{n+1}{2}$.

  • If $n$ is even, the median is the average of the two middle values at positions $\frac{n}{2}$ and $\frac{n}{2} + 1$.

• Mode

  • The value that appears most frequently in a data set.

  • A data set can be unimodal, bimodal, multimodal, or have no mode.

Special Products and Factoring

Factoring is the process of writing a polynomial as a product of simpler polynomials.

• Greatest Common Factor (GCF)

  • Factor out the highest common factor shared by all terms.

  • Pattern: $ab + ac = a(b + c)$

• Difference of Two Squares (DOTS)

  • Applicable to binomials where both terms are perfect squares separated by a minus sign.

  • Pattern: $a^2 - b^2 = (a - b)(a + b)$

• Perfect Square Trinomials (PST)

  • Trinomials resulting from squaring a binomial.

  • Pattern 1: $a^2 + 2ab + b^2 = (a + b)^2$

  • Pattern 2: $a^2 - 2ab + b^2 = (a - b)^2$

• General Quadratic Trinomials (GQT)

  • Case 1 ($a = 1$): $x^2 + bx + c = (x + m)(x + n)$, where $m + n = b$ and $m \times n = c$

  • Case 2 ($a \neq 1$): $ax^2 + bx + c$. Multiply $a \times c$, find factors that sum to $b$, rewrite the middle term, and factor by grouping.

• Factoring by Grouping

  • Used for polynomials with four terms.

  • Pattern: $ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$

Word Problems Involving Special Products and Factoring

• Translate word problems into polynomial equations, often involving geometric dimensions like area or perimeter.

• Example: Given area $A = x^2 - 9$, factor to find dimensions $(x + 3)$ and $(x - 3)$.

Operations on Rational Algebraic Expressions

A rational algebraic expression is an expression of the form $\frac{P}{Q}$, where $P$ and $Q$ are polynomials and $Q \neq 0$.

• Simplifying

  • Factor numerator and denominator completely, then cancel common factors: $\frac{a \times c}{b \times c} = \frac{a}{b}$

• Multiplication

  • Multiply numerators together and denominators together: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$

• Division

  • Multiply by the reciprocal of the divisor: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$

• Addition and Subtraction

  • Like Denominators: $\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}$

  • Unlike Denominators: Find the Least Common Denominator (LCD), convert expressions to equivalent fractions, and combine.

Rational Algebraic Equations

An equation containing one or more rational expressions.

• Steps to Solve:

  1. Determine the LCD of all terms in the equation.

  2. Multiply both sides of the equation by the LCD to eliminate fractions.

  3. Solve the resulting polynomial equation.

  4. Check for extraneous solutions that make any denominator equal to $0$.

Solving Real-World Problems Involving Rational Algebraic Equations

• Work Problems

  • Formula for combined work rate: $\frac{1}{t<em>1} + \frac{1}{t</em>2} = \frac{1}{t_{\text{total}}}$

• Uniform Motion Problems

  • Formula: $\text{Time} = \frac{\text{Distance}}{\text{Rate}}$ or $t = \frac{d}{r}$

  • Express relationships as rational equations to solve for unknown variables like speed or duration.