[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-493584-105":59,"doc-detail-493584-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","polynomial-equations-and-factoring-chapter-7-polynomial-equations-and-factoring","Polynomial Equations and Factoring - Chapter 7 - Polynomial Equations and Factoring","","Chapter 7 focuses on polynomial equations and factoring, beginning with operations such as adding and subtracting polynomials and extending to special products and solving polynomial equations in factored form. It includes practice on simplifying algebraic expressions using properties like distributive and commutative laws. The chapter also develops methods for finding the greatest common factor through prime factorizations, then applies those ideas to factor polynomials completely. It further introduces algebra tiles and model-based reasoning to write expressions and support learning checks.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/polynomial-equations-and-factoring-chapter-7-polynomial-equations-and-factoring/493584/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/polynomial-equations-and-factoring-chapter-7-polynomial-equations-and-factoring/493584.png","ImageObject",300,407,{"name":92,"@type":93},"SundayMiles","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-10-01",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"How are polynomials added and subtracted in this chapter?","Question",{"text":113,"@type":114},"Addition and subtraction are carried out by combining like terms after distributing signs appropriately. The chapter emphasizes writing the modeled expressions step by step.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"What method does the chapter use to find the greatest common factor (GCF)?",{"text":118,"@type":114},"It uses prime factorization for each number, then multiplies the common prime factors to get the GCF.",{"name":120,"@type":111,"acceptedAnswer":121},"How do algebra tiles help students learn algebraic expressions?",{"text":122,"@type":114},"Algebra tiles provide concrete models for variables and their operations, helping students translate visual arrangements into algebraic expressions and operations.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},493584,1791262833,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":140,"language":141,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":142,"faqs":143,"seo_title":144,"seo_description":67,"update_tm":145,"read_time":146,"preview_page_count":4},1099529839202,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","7  \n7.1  \n7.2  \n7.3  \n7.4  \n7.5  \n7.6  \n7.7  \n7.8  \nPolynomial Equations and Factoring  \nAdding and Subtracting Polynomials Multiplying Polynomials  \nSpecial Products of Polynomials  \nSolving Polynomial Equations in Factored Form Factoring x2 + bx + c  \nFactoring ax2 + bx + c Factoring Special Products  \nFactoring Polynomials Completely  \nFraming a Photo (p. 370)  \nGame Reserve  (p. 394)  \nMaintaining Mathematical Proficiency  \nSimplifying Algebraic Expressions  \nExample 1 Simplify 6x + 5 − 3x − 4.  \n6x + 5 − 3x − 4 = 6x − 3x + 5 − 4 Commutative Property of Addition  \n= (6 − 3)x + 5 − 4 Distributive Property = 3x + 1 Simplify.  \nExample 2 Simplify −8(y − 3) + 2y.  \n−8(y − 3) + 2y = −8(y) − (−8)(3) + 2y Distributive Property  \n= −8y + 24 + 2y Multiply.  \n= −8y + 2y + 24 Commutative Property of Addition  \n= (−8 + 2)y + 24 Distributive Property = −6y + 24 Simplify.  \nSimplify the expression.  \n1. 3x − 7 + 2x 2. 4r + 6 − 9r − 1 3. −5t + 3 − t − 4 + 8t  \n4. 3(s − 1) + 5 5. 2m − 7(3 − m) 6. 4(h + 6) − (h − 2)  \nFinding the Greatest Common Factor  \nExample 3 Find the greatest common factor (GCF) of 42 and 70.  \nTo find the GCF of two numbers, first write the prime factorization of each number. Then find the product of the common prime factors.  \n42 = 2 ⋅3 ⋅7  \n70 = 2 ⋅5 ⋅7  \nThe GCF of 42 and 70 is 2 ⋅7 = 14.  \nFind the greatest common factor.  \n7. 20, 36 8. 42, 63 9. 54, 81 10. 72, 84 11. 28, 64 12. 30, 77  \n13. ABSTRACT REASONING Is it possible for two integers to have no common factors? Explain your reasoning.  \nDynamic Solutions available [at BigIdeasMath.com](at BigIdeasMath.com)  \n355  \nMathematical Practices  \nUsing Models  \nMathematically proficient students consider concrete models when solving a mathematics problem.  \nCore Concept  \nUsing Algebra Tiles  \nWhen solving a problem, it can be helpful to use a model. For instance, you can use algebra tiles to model algebraic expressions and operations with algebraic expressions.  \n1 −1  \nx −x  \nx 2 −x2  \nWriting Expressions Modeled by Algebra Tiles  \nWrite the algebraic expression modeled by the algebra tiles.  \nSOLUTION  \na. The algebraic expression is x2.  \nb. The algebraic expression is 3x + 4.  \nc. The algebraic expression is x2 − x + 2.  \nMonitoring Progress  \nWrite the algebraic expression modeled by the algebra tiles.  \n356 Chapter 7 Polynomial Equations and Factoring  \n7.1 Adding and Subtracting Polynomials  \nREASONING ABSTRACTLY  \nEssential Question How can you add and subtract polynomials?  \n Adding Polynomials  \nWork with a partner. Write the expression modeled by the algebra tiles in each step.  \nStep 1  \nStep 2  \nStep 3  \nStep 4  \n+  \n(3x + 2) + (x − 5)  \nSubtracting Polynomials  \nWork with a partner. Write the expression modeled by the algebra tiles in each step.  \nStep 1  \nStep 2  \nStep 3  \nStep 4  \nStep 5  \n(x2 + 2x + 2) − (x − 1)  \nCommunicate Your Answer  \n3. How can you add and subtract polynomials?  \n4. Use your methods in Question 3 to find each sum or difference.  \na. (x2 + 2x − 1) + (2x2 − 2x + 1) b. (4x + 3) + (x − 2)  \nc. (x2 + 2) − (3x2 + 2x + 5) d. (2x − 3x) − (x2 − 2x + 4)  \nSection 7.1 Adding and Subtracting Polynomials 357  \n7.1 Lesson  \n Core Vocabulary  monomial, p. 358  \ndegree of a monomial, p. 358 polynomial, p. 359  \nbinomial, p. 359  \ntrinomial, p. 359  \ndegree of a polynomial, p. 359 standard form, p. 359  \nleading coefficient, p. 359 closed, p. 360  \nWhat You Will Learn  \n Find the degrees of monomials.  \n Classify polynomials.  \n Add and subtract polynomials.  \n Solve real-life problems.  \nFinding the Degrees of Monomials  \nA monomial is a number, a variable, or the product of a number and one or more variables with whole number exponents.  \nThe degree of a monomial is the sum of the exponents of the variables in the monomial. The degree of a nonzero constant term is 0. The constant 0 does not have a degree.  \n\n| Monomial | Degree |\n| --- | --- |\n| 10 | 0 |\n| 3x | 1 |\n| ab2 | 1 + 2 = 3 |\n| −1.8m5 | 5 |\n\n| 5 + x | A sum is not a monomial. |\n","cbCaiqqvk9l4VRWW","https://ap.wps.com/l/cbCaiqqvk9l4VRWW","pdf",14528139,62,"English","# Polynomial Equations and Factoring\n## 7.1 Adding and Subtracting Polynomials\n## Finding the Greatest Common Factor\n## Simplifying Algebraic Expressions\n## Using Models: Algebra Tiles\n## Finding the Degrees of Monomials\n## Classifying Polynomials","[{\"question\":\"How are polynomials added and subtracted in this chapter?\",\"answer\":\"Addition and subtraction are carried out by combining like terms after distributing signs appropriately. The chapter emphasizes writing the modeled expressions step by step.\"},{\"question\":\"What method does the chapter use to find the greatest common factor (GCF)?\",\"answer\":\"It uses prime factorization for each number, then multiplies the common prime factors to get the GCF.\"},{\"question\":\"How do algebra tiles help students learn algebraic expressions?\",\"answer\":\"Algebra tiles provide concrete models for variables and their operations, helping students translate visual arrangements into algebraic expressions and operations.\"}]","Polynomial Equations and Factoring - Chapter 7 - Polynomial Equations and Factoring | PDF",1790884242,156]