Algebra 2: Rational Functions Domain

• Algebra 2 builds upon the fundamentals of algebra, extending concepts to more complex equations, functions, and problem-solving techniques.
• Rational functions, simplifying rational functions, multiplying rational expressions, are key topics.
• Complex fractions and synthetic division are also key topics.

Domain of a Rational Function

• A rational function is defined by a rational fraction, where both the numerator and denominator are polynomials.
• The function is generally expressed as f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomial functions, and Q(x) ≠ 0.
• The domain includes all real numbers, except for x-values that make the denominator, Q(x), zero.
• Find the domain by setting Q(x) = 0 and solving for x; these x-values must be excluded.
• Express the domain in interval notation, excluding the x-values that result in a zero denominator.
• For example, if f(x) = (x+1) / (x-2), the domain includes all real numbers except x = 2, written as (-∞, 2) U (2, ∞).
• If the denominator has no real roots, such as x² + 1, the function's domain includes all real numbers, written as (-∞, ∞).
• Identifying the domain is essential for determining where the function is defined and undefined, especially regarding vertical asymptotes or holes.

Simplifying Rational Functions

• Simplification involves reducing the fraction to its simplest form by factoring the numerator and denominator, then canceling common factors.
• Begin by factoring both the numerator and the denominator into their respective factors.
• Identify and cancel any factors that are common to both the numerator and the denominator.
• The simplified form results after all possible common factors have been canceled.
• Note any restrictions on the variable, which include values that make the original denominator zero, even if they do not make the simplified denominator zero.
• For example, simplifying (x² - 4) / (x² + 4x + 4) involves factoring to ((x - 2)(x + 2)) / ((x + 2)(x + 2)), canceling (x + 2) to get (x - 2) / (x + 2), with the restriction x ≠ -2.
• Simplifying makes it easier to analyze behavior, determine asymptotes, and solve related equations.
• Always look for factoring opportunities using methods like difference of squares, perfect square trinomials, or simple trinomial factoring.

Multiplying Rational Expressions

• Multiplying rational expressions mirrors the multiplication of numerical fractions.
• Multiply the numerators to obtain the new numerator and the denominators to obtain the new denominator.
• Before multiplying, factor each numerator and denominator to identify common factors for cancellation.
• After multiplying, simplify the resulting expression by canceling any common factors between the numerator and denominator.
• Note restrictions on the variable from all denominators involved, including those canceled during simplification.
• For example, (x/y) * (y²/z) = (x * y²) / (y * z) = xy/z, restricted by y ≠ 0 and z ≠ 0.
• This multiplication is a fundamental operation in algebraic manipulations and is used in more complex problems.
• Proper factoring and simplification are crucial for achieving the correct result and identifying all variable restrictions.

Complex Fractions

• A complex fraction contains fractions in its numerator, denominator, or both.
• Simplify a complex fraction by first simplifying the numerator and denominator separately, if possible.
• Method 1 involves finding a common denominator for all fractions within the complex fraction, combining them into single fractions in both the numerator and denominator, achieved by multiplying the top and bottom by the common denominator.
• Method 2 involves rewriting the complex fraction as a division problem, then multiplying by the reciprocal of the denominator.
• Simplify to a single fraction, then reduce it to its simplest form by canceling common factors.
• Note restrictions on the variable from all denominators within the complex fraction.
• For example, simplifying (1/x + 1) / (1 - 1/x) involves simplifying the numerator to (1+x)/x and the denominator to (x-1)/x, then dividing: ((1+x)/x) / ((x-1)/x) = (1+x)/(x-1), with restrictions x ≠ 0 and x ≠ 1.
• These fractions often appear in calculus and advanced algebra, making their simplification essential.
• Complex fractions can be simplified using different approaches depending on the structure of the fraction.

Synthetic Division

• Synthetic division is a shortcut for dividing a polynomial by a linear factor in the form (x - k).
• List the polynomial's coefficients in order, using zeros for any missing terms.
• Write the value of k (from the divisor x - k) on the left.
• Bring down the first coefficient, multiply it by k, and write the result under the next coefficient.
• Add the second coefficient and the result from the previous step, repeating until all coefficients are used.
• The last number represents the remainder, while the others are the coefficients of the quotient; the quotient's degree is one less than the original polynomial's degree.
• If the remainder is zero, then (x - k) is a factor of the polynomial.
• For instance, dividing (x³ - 4x² + 6x - 4) by (x - 2) involves using coefficients 1, -4, 6, -4 and k = 2, yielding a quotient of x² - 2x + 2 with no remainder.
• Synthetic division is useful for finding roots and factoring polynomials.
• It is a fast and efficient method for polynomial division by a linear factor.

Understand rational functions in Algebra 2. Learn how to define rational functions, identify their general form as f(x) = P(x) / Q(x), and determine their domain by excluding values that make the denominator zero. Express the domain in interval notation.