Glossary
Complete Factorization
The process of factoring a polynomial until no more factors can be extracted, meaning all factors are irreducible over the integers.
Example:
The complete factorization of 2x² - 8 is 2(x - 2)(x + 2), not just 2(x² - 4).
Complex Numbers
Numbers that can be expressed in the form *a + bi*, where *a* and *b* are real numbers, and *i* is the imaginary unit (√-1).
Example:
The roots of x² + 1 = 0 are i and -i, which are complex numbers.
Degree (of a Polynomial)
The highest exponent of the variable in a polynomial, which also indicates the maximum number of solutions the polynomial can have.
Example:
The polynomial 4x³ - 2x + 1 has a degree of 3.
Difference of Cubes
A special product pattern of the form *a³ - b³* that factors into *(a - b)(a² + ab + b²)*.
Example:
y³ - 27 is a difference of cubes, factoring to (y - 3)(y² + 3y + 9).
Difference of Squares
A special product pattern of the form *a² - b²* that always factors into *(a + b)(a - b)*.
Example:
x² - 25 is a difference of squares, factoring to (x + 5)(x - 5).
Distributive Property
A fundamental property used to multiply a term by each term inside a set of parentheses, or each term in one polynomial by every term in another.
Example:
Using the distributive property, 2x(x + 3) becomes 2x² + 6x.
FOIL Method
A mnemonic (First, Outer, Inner, Last) used as a systematic way to multiply two binomials.
Example:
To multiply (x + 2)(x - 3), you use FOIL to ensure all terms are multiplied: x² - 3x + 2x - 6.
Factoring (Polynomials)
The process of breaking down a polynomial into a product of simpler expressions, called factors.
Example:
Factoring the polynomial x² - 4 yields (x - 2)(x + 2).
Factoring Trinomials
The process of breaking down a three-term polynomial, typically of the form *ax² + bx + c*, into two binomial factors.
Example:
Factoring the trinomial x² + 5x + 6 results in (x + 2)(x + 3).
Factoring by Grouping
A technique used for polynomials with four or more terms, where terms are strategically grouped to find common factors and then factor further.
Example:
To factor x³ + 2x² + 3x + 6, you can use factoring by grouping to get x²(x + 2) + 3(x + 2).
Greatest Common Factor (GCF)
The largest factor that divides all terms in a polynomial, which is then extracted to simplify the expression.
Example:
The GCF of 6x² + 9x is 3x, so it factors to 3x(2x + 3).
Like Terms
Terms within a polynomial that have the same variables raised to the same powers, allowing them to be combined through addition or subtraction.
Example:
In the expression 5x² + 2x² - 3x, 5x² and 2x² are like terms.
Number of Solutions (of a Polynomial)
The maximum count of roots a polynomial can have, which is always equal to its degree.
Example:
A polynomial with a degree of 4 can have at most four solutions (counting multiplicity and complex solutions).
Polynomials
Algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, multiplication, and non-negative integer exponents.
Example:
The expression 3x² - 5x + 7 is a polynomial.
Roots/Zeros (of a Polynomial)
The values of the variable that make a polynomial equal to zero; these are the solutions to a polynomial equation.
Example:
For the equation x² - 4 = 0, the roots (or zeros) are x = 2 and x = -2.
Standard Form (of a Polynomial)
A polynomial arranged with its terms ordered from the highest degree to the lowest degree.
Example:
The polynomial 7 - 2x + 5x² in standard form is 5x² - 2x + 7.
Sum of Cubes
A special product pattern of the form *a³ + b³* that factors into *(a + b)(a² - ab + b²)*.
Example:
x³ + 8 is a sum of cubes, factoring to (x + 2)(x² - 2x + 4).
Zero Product Property
A fundamental principle stating that if the product of two or more factors is zero, then at least one of those factors must be zero.
Example:
If (x - 3)(x + 5) = 0, then by the zero product property, x - 3 = 0 or x + 5 = 0.