4: Factoring Techniques
- Page ID
- 173413
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- 4.1: Greatest Common Factor
- This page introduces the concept of factoring polynomials, primarily focusing on the Greatest Common Factor (GCF). It explains its definition as the largest expression that divides multiple expressions and stresses the connection between factoring and the Distributive Property. Through examples, the page illustrates how to identify and extract both constants and variable factors, including negative coefficients.
- 4.2: Factoring by Grouping
- This page explains how to factor polynomials with four terms through the method of factoring by grouping, utilizing the Grouping Identity theorem. It includes detailed steps, examples, and tips for identifying common binomial factors, addressing negatives, and rearranging terms. The emphasis is on matching binomial factors and being cautious with signs during the factoring process, providing a comprehensive guide for effective polynomial factorization.
- 4.3: Factoring Trinomials
- This page explains the process of factoring quadratic trinomials, detailing definitions, relevant theorems, and techniques such as guess-and-check. Key strategies include identifying factor signs, extracting the greatest common factor, and verifying results through expansion. It also covers determining if trinomials can be factored over integers, emphasizing the need to find integer pairs for specific criteria.
- 4.4: Factoring Differences of Squares
- This page explains factoring polynomials, particularly the difference of squares, by defining perfect squares and illustrating that \(a^2 - b^2\) factors to \((a - b)(a + b)\). It stresses the recognition of these patterns and notes that sums of squares cannot be factored over real numbers. Examples are provided to show how to identify and apply the difference of squares, including the need to factor out any greatest common factor beforehand.
- 4.5: Factoring Sums and Differences of Cubes
- This page covers the factoring of the sum and difference of cubes in algebra, explaining perfect cubes and providing relevant formulas. It introduces the SOAP mnemonic to remember sign patterns in factorizations and emphasizes that resulting trinomial factors are irreducible over the reals. Additionally, it includes examples to demonstrate these concepts and stresses the importance of identifying perfect cubes and factoring out the greatest common factor.
- 4.6: Factoring Expressions Using the Rational Roots Theorem
- This page explains the Rational Root Theorem, which identifies potential rational roots of polynomials in the form \(\pm \frac{p}{q}\) (where \(p\) and \(q\) are factors of the constant and leading coefficients, respectively). It outlines the steps for finding candidates using synthetic division, and discusses the Remainder and Factor Theorems.
- 4.7: Factoring Expressions Involving Negative Rational Exponents
- This page explains negative and rational exponents, detailing their definitions and handling in expressions. It clarifies that negative exponents indicate reciprocals while rational exponents relate to roots and powers. The text also presents a theorem for factoring expressions with a common base and offers practical tips for identifying common factors and ensuring nonnegative exponents, accompanied by illustrative examples.
- 4.8: Factoring Expressions that are Quadratic-in-Form
- This page explains how to transform non-quadratic expressions into quadratic forms using \(u\)-substitution. It defines quadratic-in-form, details the exponent relationships, and outlines the factoring process through substitution. The page provides several examples to illustrate identifying and factoring these expressions, effectively guiding the reader through complete factorization using the original expressions.