Glossary

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 $\frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{4}}$ is a complex fraction that needs to be simplified.

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

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

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

Grouping

Criticality: 1

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

Example:

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

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

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

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 $\frac{x^2 - 4}{x+2}$ 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 $\frac{x+1}{x-3}$ , the restrictions on the variable are $x \neq 3$ , because if $x=3$ , the denominator would be zero.

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 $\frac{5x-1}{(x-1)(x+2)}$ as $\frac{A}{x-1} + \frac{B}{x+2}$ .

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

Glossary

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 $\frac{\frac{1}{2} + \frac{1}{3}}{\frac{1}{4}}$ is a complex fraction that needs to be simplified.

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

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

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

Grouping

Criticality: 1

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

Example:

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

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

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

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 $\frac{x^2 - 4}{x+2}$ 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 $\frac{x+1}{x-3}$ , the restrictions on the variable are $x \neq 3$ , because if $x=3$ , the denominator would be zero.

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 $\frac{5x-1}{(x-1)(x+2)}$ as $\frac{A}{x-1} + \frac{B}{x+2}$ .

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