Glossary
AC Method
A technique for factoring quadratic expressions $ax^2 + bx + c$ by finding two numbers that multiply to 'ac' and add to 'b', then rewriting the middle term.
Example:
To factor using the AC Method, you'd look for two numbers that multiply to and add to 7 (which are 1 and 6).
Completing the Square
A method used to rewrite a quadratic expression in the form $(x + p)^2 + q$, which is particularly useful for finding the vertex of a parabola or solving quadratic equations.
Example:
Rewriting as is an example of completing the square.
Difference of Cubes
A special polynomial form $a^3 - b^3$ that factors into $(a - b)(a^2 + ab + b^2)$.
Example:
To factor , you would apply the difference of cubes formula, yielding .
Difference of Squares
A special polynomial form $a^2 - b^2$ that always factors into $(a + b)(a - b)$. Recognizing this pattern simplifies factoring significantly.
Example:
The expression is a difference of squares because it can be written as , factoring to .
Factoring
The process of rewriting a polynomial as a product of its factors, essentially reverse multiplication. It's crucial for simplifying expressions and solving equations.
Example:
When you break down into , you are factoring the quadratic expression.
Factoring by Grouping
A method used to factor polynomials with four or more terms by grouping terms with common factors and then factoring out a common binomial.
Example:
To factor , you might use factoring by grouping to get , leading to .
Perfect Square Trinomials
Trinomials that result from squaring a binomial, taking the form $a^2 + 2ab + b^2 = (a + b)^2$ or $a^2 - 2ab + b^2 = (a - b)^2$.
Example:
is a perfect square trinomial because it factors directly to .
Quadratic Equations
Equations of the form $ax^2 + bx + c = 0$, where 'a' is not zero. Factoring is a primary method for solving these equations.
Example:
Solving means finding the values of that satisfy this quadratic equation.
Quadratic Expression
A polynomial of degree two, typically in the form $ax^2 + bx + c$, where 'a' is not zero. Factoring these expressions is a fundamental SAT skill.
Example:
The expression is a quadratic expression that can often be factored.
Sum of Cubes
A special polynomial form $a^3 + b^3$ that factors into $(a + b)(a^2 - ab + b^2)$.
Example:
Factoring involves the sum of cubes formula, resulting in .
Zero Product Property
A fundamental property stating that if the product of two or more factors is zero, then at least one of the factors must be zero. This is key to solving factored equations.
Example:
If , the Zero Product Property tells us that either or , leading to solutions or .