Glossary

10.6 Introduction to Factoring Polynomials

2 min read•june 25, 2024

polynomials is a key skill in algebra. It's all about breaking down complex expressions into simpler parts. This makes solving equations and simplifying expressions much easier.

Learning to factor helps you understand the structure of polynomials. You'll use techniques like finding common factors, grouping terms, and working with special patterns. These skills are crucial for more advanced math topics.

Factoring Polynomials

Greatest common factor identification

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  • Largest factor that divides all terms in an algebraic expression without a remainder
  • Found by listing factors of coefficients, identifying the largest number that is a factor of all coefficients
  • List variables appearing in every term, find the lowest for each
  • is the product of the GCF of coefficients and common variables raised to their lowest exponents
  • Essential for simplifying expressions and solving equations by factoring

Factoring out common factors

  • Rewriting a as a product of factors by dividing each term by the GCF
  • Steps: identify GCF of all terms, divide each term by GCF, write factored expression as product of GCF and quotient
  • Example: 10x3+15x210x^3 + 15x^2 factored is 5x2(2x+3)5x^2(2x + 3) because 5x25x^2 is the GCF
  • Useful for simplifying complex polynomials and solving equations by factoring

Techniques for polynomial factoring

  • : grouping terms, factoring out common factors
    • Example: ax+ay+bx+by=a(x+y)+b(x+y)=(a+b)(x+y)ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)
  • Factoring trinomials using trial and error or decomposition
    • Sum/: a3±b3=(a±b)(a2∓ab+b2)a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)
    • Example: x2+5x+6=(x+2)(x+3)x^2 + 5x + 6 = (x + 2)(x + 3)
  • Factoring four-term polynomials by grouping or substitution
    • Example: 2x3+3x2−8x−12=(2x2−8)(x+2)2x^3 + 3x^2 - 8x - 12 = (2x^2 - 8)(x + 2)
  • Applications: simplifying expressions, solving equations, finding /, analyzing polynomial functions
  • : a shortcut method for dividing polynomials by linear factors
  • : helps identify potential rational roots of a polynomial equation
  • : relates the roots of a polynomial to its factors
  • : used to solve quadratic equations when factoring is not possible
    • The in the quadratic formula determines the nature of the roots

Key Terms to Review (21)

Binomial: A binomial is a polynomial expression with exactly two terms. It consists of two monomials, typically with different variables or exponents, connected by an operation such as addition, subtraction, multiplication, or division.
Coefficient: A coefficient is a numerical factor that multiplies a variable in an algebraic expression. It represents the scale or magnitude of the variable, indicating how much of that variable is present in the expression.
Difference of Cubes: The difference of cubes is a special type of polynomial expression that can be factored using a specific formula. It refers to the difference between two perfect cubes, such as $a^3 - b^3$, where $a$ and $b$ are variables or numbers.
Discriminant: The discriminant is a mathematical expression that provides information about the nature and number of solutions to a quadratic equation. It is a key concept in the factorization of polynomials, particularly those of degree two or higher.
Exponent: An exponent is a mathematical symbol that indicates the number of times a base number is multiplied by itself. It represents repeated multiplication and is used to express large numbers concisely. Exponents are a fundamental concept in algebra and are crucial for understanding and working with expressions, polynomials, and scientific notation.
Factor Theorem: The Factor Theorem is a fundamental principle in polynomial factorization. It states that a polynomial $P(x)$ is divisible by $(x-a)$ if and only if $P(a) = 0$. In other words, the factor $(x-a)$ is a factor of the polynomial $P(x)$ if and only if the polynomial evaluates to 0 when $x$ is replaced by $a$.
Factoring: Factoring is the process of breaking down a polynomial expression into a product of smaller, simpler expressions. It involves identifying common factors among the terms and expressing the polynomial as a product of those factors.
Factoring by Grouping: Factoring by grouping is a technique used to factor polynomials by identifying common factors among groups of terms within the polynomial. This method helps simplify the factorization process by breaking down the polynomial into smaller, more manageable parts.
Factoring Out Common Factors: Factoring out common factors is a technique in algebra where the greatest common factor (GCF) of a polynomial expression is identified and extracted, resulting in a simplified expression. This process helps to break down complex polynomial expressions into more manageable components.
Four-Term Polynomial: A four-term polynomial is an algebraic expression consisting of four terms, each of which is a product of a coefficient and one or more variables raised to a power. These polynomials are an important focus in the study of factoring techniques, as they often require specific strategies to identify their factors.
GCF: GCF, or Greatest Common Factor, is a fundamental concept in mathematics that refers to the largest positive integer that divides two or more integers without a remainder. The GCF is crucial in understanding prime factorization and factoring polynomials, as it helps simplify expressions and find common factors among numbers or algebraic terms.
Greatest Common Factor: The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. It is an essential concept in number theory and is closely related to finding multiples, prime factorization, and factoring polynomials.
Polynomial: A polynomial is an algebraic expression that consists of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are fundamental to understanding and working with algebraic expressions, as they form the building blocks for many mathematical concepts and applications.
Quadratic Formula: The quadratic formula is a mathematical equation used to solve quadratic equations, which are polynomial equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers. The formula provides a systematic way to find the roots or solutions of a quadratic equation.
Rational Root Theorem: The Rational Root Theorem is a mathematical principle that helps determine the possible rational roots (also known as polynomial roots) of a polynomial equation. It states that if a polynomial equation with integer coefficients has a rational root p/q, where p is an integer and q is a positive integer, then p must be a factor of the constant term and q must be a factor of the leading coefficient.
Roots: In the context of factoring polynomials, the term 'roots' refers to the values of the variable(s) that make the polynomial equation equal to zero. Roots are the solutions to the polynomial equation, and they are crucial in understanding the behavior and properties of polynomial functions.
Sum of Cubes: The sum of cubes is the sum of the cubes of two or more numbers. It is a special case of polynomial addition and is an important concept in factoring polynomials.
Synthetic Division: Synthetic division is a shortcut method for dividing polynomials that simplifies the division process. It allows you to find the quotient and remainder of a polynomial division without having to carry out the full long division algorithm.
Trinomial: A trinomial is a polynomial expression with three terms. It is a type of polynomial that can be represented in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Trinomials are fundamental in the study of polynomials and play a crucial role in various algebraic operations and factorization techniques.
Variable: A variable is a symbol, typically a letter, that represents an unknown or changeable quantity in an algebraic expression or equation. It is a fundamental concept in algebra that allows for the generalization of mathematical relationships and the solution of problems involving unknown values.
Zeros: Zeros, in the context of polynomials, refer to the values of the variable(s) that make the polynomial expression equal to zero. They represent the points where the graph of the polynomial intersects the x-axis, providing valuable information about the behavior and factorization of the polynomial.
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