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Glossary

C

Complex Fractions

Criticality: 2

Fractions that contain other fractions in their numerator, denominator, or both. They are simplified by multiplying by the LCD of all internal fractions.

Example:

The expression 12+1314\frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{4}} is a complex fraction that needs to be simplified.

D

Difference of Squares

Criticality: 2

A specific factoring pattern for binomials in the form $a^2 - b^2$, which factors into $(a+b)(a-b)$.

Example:

Recognizing x2−25x^2 - 25 as a Difference of Squares allows you to quickly factor it into (x+5)(x−5)(x+5)(x-5).

F

Factoring

Criticality: 3

The process of breaking down a polynomial into a product of simpler polynomials (its factors). It's crucial for simplifying rational expressions.

Example:

To simplify x2−9x−3\frac{x^2 - 9}{x-3}, you first need to factor the numerator into (x−3)(x+3)(x-3)(x+3).

G

Greatest Common Factor (GCF)

Criticality: 2

The largest factor that two or more terms or polynomials share. Pulling out the GCF is often the first step in factoring.

Example:

In the expression 3x2+6x3x^2 + 6x, the Greatest Common Factor is 3x3x, allowing you to factor it as 3x(x+2)3x(x+2).

Grouping

Criticality: 1

A factoring technique used for polynomials with four terms, where terms are grouped to find common binomial factors.

Example:

To factor x3+2x2+3x+6x^3 + 2x^2 + 3x + 6, you can use grouping to get x2(x+2)+3(x+2)x^2(x+2) + 3(x+2), which simplifies to (x2+3)(x+2)(x^2+3)(x+2).

L

Least Common Denominator (LCD)

Criticality: 3

The smallest polynomial that is a multiple of all the denominators in a set of rational expressions. It's essential for adding and subtracting fractions.

Example:

To add 1x+1x+1\frac{1}{x} + \frac{1}{x+1}, the Least Common Denominator is x(x+1)x(x+1).

Long Division (for rational expressions)

Criticality: 2

An algebraic method used to divide polynomials, especially when the degree of the numerator is greater than or equal to the degree of the denominator.

Example:

When simplifying x2+5x+6x+1\frac{x^2 + 5x + 6}{x+1}, you might use Long Division to find the quotient and remainder.

R

Rational Expressions

Criticality: 3

Fractions where both the numerator and the denominator are polynomials. They combine concepts of factoring, fractions, and algebraic manipulation.

Example:

The expression x2−4x+2\frac{x^2 - 4}{x+2} is a rational expression that can be simplified.

Restrictions on the variable

Criticality: 3

Values of the variable that would make the denominator of a rational expression equal to zero, thus making the expression undefined. These values must be excluded from the domain.

Example:

For the expression x+1x−3\frac{x+1}{x-3}, the restrictions on the variable are x≠3x \neq 3, because if x=3x=3, the denominator would be zero.

S

Splitting Fractions

Criticality: 1

A technique, often called partial fraction decomposition, used to break down a complex rational expression into a sum of simpler fractions with simpler denominators.

Example:

To integrate certain rational functions in calculus, you might use splitting fractions to rewrite 5x−1(x−1)(x+2)\frac{5x-1}{(x-1)(x+2)} as Ax−1+Bx+2\frac{A}{x-1} + \frac{B}{x+2}.

Sum or Difference of Cubes

Criticality: 1

Specific factoring patterns for binomials in the form $a^3 + b^3$ or $a^3 - b^3$, which factor into $(a+b)(a^2-ab+b^2)$ and $(a-b)(a^2+ab+b^2)$ respectively.

Example:

Factoring x3−8x^3 - 8 requires applying the Difference of Cubes formula to get (x−2)(x2+2x+4)(x-2)(x^2+2x+4).