Intermediate Algebra
Unit 6 Overview: Factoring
6.1 Greatest Common Factor and Factor by Grouping
6.2 Factor Trinomials
6.3 Factor Special Products
6.4 General Strategy for Factoring Polynomials
6.5 Polynomial Equations

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6.2 Factor Trinomials

Last Updated on June 24, 2024

Factoring trinomials is a key skill in algebra. It's about breaking down complex expressions into simpler parts, making equations easier to solve. This process helps us understand the roots of quadratic equations and simplify algebraic fractions.

There are several methods for factoring trinomials, each suited to different forms. The product-sum method works for simpler trinomials, while the ac method tackles more complex ones. Substitution can simplify tricky expressions before factoring.

Factoring Trinomials

Product-sum method for trinomials

  • Factors trinomials in the form x2+bx+cx^2 + bx + c (a quadratic expression)
    • Find two numbers that multiply to give cc and add to give bb
    • Rewrite the trinomial as x2+mx+nx+cx^2 + mx + nx + c (m+n=bm + n = b and mn=cmn = c)
    • Factor by grouping: x(x+m)+n(x+m)x(x + m) + n(x + m)
    • Factor out the common binomial: (x+m)(x+n)(x + m)(x + n) (these are binomial factors)
  • Example: Factor x2+7x+12x^2 + 7x + 12
    • Numbers that multiply to give 12 and add to give 7 are 3 and 4
    • Rewrite as x2+3x+4x+12x^2 + 3x + 4x + 12
    • Factor by grouping: x(x+3)+4(x+3)x(x + 3) + 4(x + 3)
    • Factor out common binomial: (x+3)(x+4)(x + 3)(x + 4)

Ac method for complex trinomials

  • Factors trinomials in the form ax2+bx+cax^2 + bx + c (a≠1a \neq 1)
    • Multiply aa and cc to get acac
    • Find two numbers that multiply to give acac and add to give bb
    • Rewrite trinomial as ax2+mx+nx+cax^2 + mx + nx + c (m+n=bm + n = b and mn=acmn = ac)
    • Factor by grouping: x(ax+m)+n(ax+m)x(ax + m) + n(ax + m)
    • Factor out common binomial and simplify: 1a(ax+m)(ax+n)\frac{1}{a}(ax + m)(ax + n)
  • Example: Factor 2x2+7x+32x^2 + 7x + 3
    • ac=2×3=6ac = 2 \times 3 = 6
    • Numbers that multiply to give 6 and add to give 7 are 6 and 1
    • Rewrite as 2x2+6x+x+32x^2 + 6x + x + 3
    • Factor by grouping: x(2x+6)+1(2x+6)x(2x + 6) + 1(2x + 6)
    • Factor out common binomial and simplify: 12(2x+6)(2x+1)\frac{1}{2}(2x + 6)(2x + 1)

Substitution in trinomial factoring

  • Simplifies and factors complex trinomial expressions
    • Identify common term or expression to substitute with a single variable
    • Rewrite trinomial using substituted variable
    • Factor simplified trinomial using appropriate method
    • Replace substituted variable with original term or expression
  • Example: Factor 9x2−12x+49x^2 - 12x + 4
    • Substitute u=3xu = 3x, trinomial becomes u2−4u+4u^2 - 4u + 4
    • Factor simplified trinomial: (u−2)2(u - 2)^2
    • Replace uu with 3x3x: (3x−2)2(3x - 2)^2

Efficient factoring technique selection

  • Identify trinomial form to determine most efficient factoring technique
    1. x2+bx+cx^2 + bx + c form: use product-sum method
    • Apply the distributive property to check your work
    1. ax2+bx+cax^2 + bx + c form (a≠1a \neq 1): use ac method
    2. Complex terms or expressions: consider substitution to simplify before factoring
  • Apply chosen factoring technique to factor trinomial completely
    • Follow steps of selected method carefully
    • Ensure factored expression is equivalent to original trinomial
  • Check work by multiplying factors to confirm they equal original trinomial

Additional Factoring Concepts

  • Greatest Common Factor (GCF): Always check for a GCF before applying other factoring methods
  • Polynomial factorization: A broader term that includes factoring trinomials and other polynomial forms
  • The distributive property is crucial in both factoring and expanding polynomials

Key Terms to Review (21)

Substitution: Substitution is the process of replacing one variable or expression with another, often to simplify an algebraic expression or equation. This technique is widely used in various mathematical contexts, including using the language of algebra, factoring trinomials, and solving equations in quadratic form.
Distributive Property: The distributive property is a fundamental algebraic rule that states that the product of a number and a sum is equal to the sum of the individual products. It allows for the simplification of expressions involving multiplication and addition or subtraction.
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 and $a \neq 0$. This formula provides a systematic way to find the solutions, or roots, of a quadratic equation.
Factor by Grouping: Factor by grouping is a technique used to factor polynomial expressions by identifying common factors among groups of terms and then factoring out those common factors. This method is particularly useful when dealing with polynomials that cannot be easily factored using other methods, such as factoring by common factors or factoring trinomials.
FOIL Method: The FOIL method is a systematic approach used to multiply binomials, where FOIL stands for First, Outer, Inner, Last. It is a widely applied technique in various algebraic operations, including multiplying polynomials, factoring trinomials, and working with special products.
Constant Term: The constant term is a numerical value that does not have a variable associated with it in a polynomial expression. It is the term that remains unchanged regardless of the value assigned to the variable(s) in the expression.
Difference of Squares: The difference of squares is a special product in algebra where the result of subtracting one perfect square from another perfect square can be factored. This concept is fundamental to understanding polynomial multiplication, factoring trinomials, factoring special products, and solving polynomial equations.
Quadratic Expression: A quadratic expression is a polynomial expression of degree two, containing a variable with the highest exponent being two. These expressions are characterized by their ability to be factored or solved using various algebraic techniques, making them an essential component in the study of algebra.
Leading Coefficient: The leading coefficient of a polynomial is the numerical coefficient of the term with the highest degree. It represents the scale or magnitude of the polynomial and plays a crucial role in various polynomial operations and properties.
Trinomials: A trinomial is a polynomial expression with three terms, typically in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are coefficients. Trinomials are a fundamental concept in algebra and are central to the topic of factoring polynomial expressions.
Zero Product Property: The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This principle is fundamental in solving various algebraic equations and expressions involving polynomials, rational functions, and radicals.
Factored Form: Factored form is a way of expressing a polynomial expression by breaking it down into a product of factors. This representation can provide valuable insights into the structure and properties of the polynomial, making it a crucial concept in the context of factoring trinomials.
Greatest Common Factor (GCF): The greatest common factor (GCF) is the largest positive integer that divides each of the given integers without a remainder. It is a fundamental concept in number theory and has important applications in factoring polynomials, particularly trinomials.
Product-Sum Method: The product-sum method is a technique used to factor trinomials, where the goal is to find two factors of the constant term that, when added, give the coefficient of the linear term. This method involves manipulating the coefficients and constant term to identify the appropriate factors that satisfy the trinomial equation.
Roots: Roots refer to the values of a variable that satisfy an equation or inequality. They represent the solutions to polynomial expressions, where the roots are the x-values that make the equation or inequality equal to zero. Roots are a fundamental concept in algebra, as they are essential for understanding and solving various types of polynomial functions and equations.
Binomial Factors: Binomial factors refer to the process of factoring a trinomial expression into the product of two binomials. This is a crucial technique in the context of factoring trinomials, as it allows for the simplification and manipulation of more complex algebraic expressions.
Complex Trinomials: Complex trinomials are quadratic expressions with three terms that contain variables raised to different powers, such as $ax^2 + bx + c$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. These expressions are considered 'complex' because they involve more than one variable term.
Polynomial Factorization: Polynomial factorization is the process of breaking down a polynomial expression into a product of simpler polynomial factors. This technique is particularly useful in the context of solving polynomial equations and simplifying algebraic expressions.
Ac Method: The ac method is a specific technique used in the factoring of polynomials, particularly trinomials. It involves identifying the constant term (c) and the coefficient of the squared term (a) in order to determine the factors of the polynomial.
Reverse FOIL: Reverse FOIL is a factoring technique used to identify the factors of a quadratic expression in the form of a trinomial. It involves working backward from the given trinomial to determine the two binomial factors that, when multiplied together, result in the original expression.
Perfect Square Trinomial: A perfect square trinomial is a special type of trinomial (an algebraic expression with three terms) that can be factored as the square of a binomial. This means that a perfect square trinomial can be expressed as the square of a sum or difference of two terms.