key term - Factoring

Definition

Factoring is the process of breaking down a polynomial expression into a product of simpler polynomial expressions. This technique is widely used in various areas of mathematics, including solving equations, simplifying rational expressions, and working with quadratic functions.

5 Must Know Facts For Your Next Test

1. Factoring is essential for solving formulas for a specific variable, as it allows you to isolate the variable of interest.
2. Multiplying polynomials can be simplified by factoring the resulting expression.
3. Factoring the greatest common factor is a crucial step in the process of factoring by grouping.
4. Factoring special products, such as the difference of two squares or the sum/difference of two cubes, can be done using specific factorization techniques.
5. Factoring rational expressions is necessary for simplifying complex rational expressions and solving rational equations.

Review Questions

• Explain how factoring can be used to solve a formula for a specific variable.
  • To solve a formula for a specific variable, you need to isolate that variable. Factoring can help achieve this by breaking down the expression into simpler factors. Once the expression is factored, you can identify the variable of interest and use algebraic manipulations, such as dividing or multiplying both sides, to solve for that variable.
• Describe the role of factoring in the process of multiplying polynomials.
  • When multiplying polynomials, the resulting expression can often be simplified by factoring. By identifying common factors among the terms and factoring them out, you can rewrite the product as a simpler expression. This can be particularly useful when working with more complex polynomial expressions, as factoring can reveal patterns and relationships that make the multiplication process more efficient.
• Analyze the importance of factoring the greatest common factor in the context of factoring by grouping.
  • Factoring by grouping is a technique used to factor polynomial expressions that do not have a readily apparent common factor. The first step in this process is to identify the greatest common factor (GCF) among the terms. Factoring out the GCF is crucial, as it allows you to group the remaining terms in a way that facilitates further factorization. By isolating the GCF, you can simplify the expression and make it more manageable to factor using other methods, such as the difference of two squares or the sum/difference of two cubes.

Related terms

Greatest Common Factor (GCF):

The largest factor that is common to all the terms in a polynomial expression.

Polynomial:

An algebraic expression consisting of variables and coefficients, where the variables are only raised to non-negative integer powers.

Quadratic Equation:

A polynomial equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a \neq 0$.