Praxis Mathematics (5165)

Dimensional analysis provides a systematic method for converting between units and verifying equation consistency. Every physical quantity has dimensions expressed in terms of fundamental quantities such as length, mass, time, temperature, and electric current.

Converting units involves multiplying by conversion factors equal to 1. To convert 5 kilometers to meters, multiply by 1000 m/1 km, yielding 5000 m. Chain conversions handle multiple unit changes: converting miles per hour to meters per second requires converting miles to meters and hours to seconds. The key principle is arranging conversion factors so unwanted units cancel.

Derived units combine fundamental units through multiplication and division. Area has units of length squared, volume has units of length cubed. Speed equals distance divided by time, with units like meters per second. Density equals mass divided by volume, with units like kilograms per cubic meter. Understanding how units combine helps verify calculations and interpret results.

Significant figures indicate measurement precision. All nonzero digits are significant. Zeros between nonzero digits are significant. Leading zeros are not significant. Trailing zeros in decimal numbers are significant. In multiplication and division, results should have the same number of significant figures as the least precise factor. In addition and subtraction, results should match the least precise decimal place.

Precision refers to the closeness of repeated measurements to each other, while accuracy refers to closeness to the true value. A measurement can be precise without being accurate (consistently wrong), accurate without being precise (varying around the true value), both, or neither. Understanding this distinction is essential for interpreting data and experimental results.

Absolute error equals the measured value minus the true value. Relative error equals absolute error divided by true value, often expressed as a percentage. Measurements should report uncertainty, typically as value ± error. When combining measurements, errors propagate: for sums and differences, add absolute errors; for products and quotients, add relative errors.

Scientific notation expresses numbers as a coefficient between 1 and 10 multiplied by a power of 10. This format simplifies calculations with very large or small numbers and clarifies significant figures. To multiply in scientific notation, multiply coefficients and add exponents. To divide, divide coefficients and subtract exponents.

An algebraic expression combines numbers, variables, and operations. Terms are the parts of an expression separated by addition or subtraction. Like terms have identical variable parts and can be combined by adding coefficients. The degree of a term is the sum of exponents on its variables.

Adding and subtracting expressions involves combining like terms. Multiplying expressions uses the distributive property. The FOIL method (First, Outer, Inner, Last) applies to binomial multiplication: (a + b)(c + d) = ac + ad + bc + bd. Squaring a binomial produces a perfect square trinomial: (a + b)² = a² + 2ab + b². The difference of squares formula gives (a + b)(a - b) = a² - b².

The product rule states aᵐ × aⁿ = aᵐ⁺ⁿ. The quotient rule gives aᵐ ÷ aⁿ = aᵐ⁻ⁿ. The power rule states (aᵐ)ⁿ = aᵐⁿ. Additional rules include (ab)ⁿ = aⁿbⁿ and (a/b)ⁿ = aⁿ/bⁿ. Zero exponent gives a⁰ = 1 for a ≠ 0. Negative exponents indicate reciprocals: a⁻ⁿ = 1/aⁿ.

Radicals and rational exponents are related by a^(1/n) = ⁿ√a. More generally, a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ. To simplify radicals, factor out perfect powers: √72 = √(36 × 2) = 6√2. Rationalizing the denominator eliminates radicals from denominators by multiplying by appropriate factors.

A polynomial is an expression consisting of terms with non-negative integer exponents. The degree of a polynomial is the highest exponent among its terms. Polynomials are classified by degree: constant (0), linear (1), quadratic (2), cubic (3), quartic (4), quintic (5).

Adding polynomials combines like terms. Subtracting requires distributing the negative sign before combining. Multiplying polynomials uses distribution: multiply each term of one polynomial by each term of the other, then combine like terms. The degree of a product equals the sum of the degrees of the factors.

Factoring reverses multiplication, expressing a polynomial as a product of simpler polynomials. Always begin by factoring out the greatest common factor (GCF). Factor trinomials x² + bx + c by finding two numbers with product c and sum b. For ax² + bx + c with a ≠ 1, use factoring by grouping or the AC method. Recognize special patterns: difference of squares a² - b² = (a + b)(a - b), perfect square trinomials a² ± 2ab + b² = (a ± b)², sum and difference of cubes a³ ± b³ = (a ± b)(a² ∓ ab + b²).

Long division of polynomials follows the same algorithm as numerical long division. Arrange terms in descending order, divide leading terms, multiply and subtract, bring down the next term, and repeat. The dividend equals divisor times quotient plus remainder. Synthetic division provides a shortcut when dividing by (x - c), using only coefficients.

The Remainder Theorem states that when polynomial P(x) is divided by (x - c), the remainder equals P(c). The Factor Theorem follows: (x - c) is a factor of P(x) if and only if P(c) = 0. These theorems connect polynomial division with evaluation and root-finding.

A root or zero of a polynomial P(x) is a value c such that P(c) = 0. The Fundamental Theorem of Algebra guarantees that a polynomial of degree n has exactly n roots (counting multiplicity) in the complex numbers. Rational Root Theorem: if p/q is a rational root of polynomial with integer coefficients, then p divides the constant term and q divides the leading coefficient.