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== 10.8#h(0.6em)Factoring Quadratic Trinomials


    Consider the trinomial


    #math.equation(block: true, alt: "x squared plus 10 x plus 16")[$x^(2) + 10 x + 16$]
    Can we find two binomial factors,


    #math.equation(block: true, alt: "open parenthesis x plus a close parenthesis open parenthesis x plus b close parenthesis")[$( x + a ) ( x + b )$]
    whose product is the given trinomial? The product of the binomials is


    #math.equation(block: true, alt: "open parenthesis x plus a close parenthesis open parenthesis x plus b close parenthesis equals x squared plus open parenthesis a plus b close parenthesis x plus a b")[$( x + a ) ( x + b ) = x^(2) + ( a + b ) x + a b$]
    Thus, we are looking for two numbers, #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$], that satisfy


    #math.equation(block: true, alt: "open parenthesis x plus a close parenthesis open parenthesis x plus b close parenthesis equals x squared plus open parenthesis a plus b close parenthesis x plus a b equals x squared plus 10 x plus 16")[$( x + a ) ( x + b ) = x^(2) + ( bold(italic(a)) + bold(italic(b)) ) x + bold(italic(a)) bold(italic(b)) = x^(2) + 10 x + 16$]
    By comparing the coefficients of the terms in the two trinomials, we see that #math.equation(block: false, alt: "a plus b equals 10")[$bold(italic(a)) + bold(italic(b)) = 10$] and #math.equation(block: false, alt: "a b equals 16")[$bold(italic(a)) bold(italic(b)) = 16$]. That is, the sum of the two numbers is the coefficient of the linear term, #math.equation(block: false, alt: "10")[$10$], and their product is the constant term, #math.equation(block: false, alt: "16")[$16$].


    To find the numbers, we list all the possible integer factorizations of #math.equation(block: false, alt: "16")[$16$]:


    #math.equation(block: true, alt: "1 times 16 , 2 times 8 , and 4 times 4")[$1 ⋅ 16 , " " " " " " 2 ⋅ 8 , " " " " " " " and " " " " " " " 4 ⋅ 4$]
    We see that only one combination gives the correct linear term: #math.equation(block: false, alt: "8")[$8$] and #math.equation(block: false, alt: "2")[$2$]. These are the numbers #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$], so


    #math.equation(block: true, alt: "x squared plus 10 x plus 16 equals open parenthesis x plus 8 close parenthesis open parenthesis x plus 2 close parenthesis")[$x^(2) + 10 x + 16 = ( x + 8 ) ( x + 2 )$]
    In Example we factor quadratic trinomials in which one or more of the coefficients is negative.


    #examplebox("Example 1")[][
Factor.


        + #math.equation(block: false, alt: "x squared minus 7 x plus 12")[$x^(2) − 7 x + 12$]
+ #math.equation(block: false, alt: "x squared minus x minus 12")[$x^(2) − x − 12$]


      
      #solutionbox[
+ Find two numbers whose product is #math.equation(block: false, alt: "12")[$12$] and whose sum is #math.equation(block: false, alt: "minus 7")[$− 7$]. Because the product is positive and the sum is negative, the two numbers must both be negative. The possible factors of #math.equation(block: false, alt: "12")[$12$] are #math.equation(block: false, alt: "minus 1")[$− 1$] and #math.equation(block: false, alt: "minus 12")[$− 12$], #math.equation(block: false, alt: "minus 2")[$− 2$] and #math.equation(block: false, alt: "minus 6")[$− 6$], or #math.equation(block: false, alt: "minus 3")[$− 3$] and #math.equation(block: false, alt: "minus 4")[$− 4$]. Only #math.equation(block: false, alt: "minus 4")[$− 4$] and #math.equation(block: false, alt: "minus 3")[$− 3$] have the correct sum, #math.equation(block: false, alt: "minus 7")[$− 7$]. Hence,
                #math.equation(block: true, alt: "x squared minus 7 x plus 12 equals open parenthesis x minus 4 close parenthesis open parenthesis x minus 3 close parenthesis")[$x^(2) − 7 x + 12 = ( x − 4 ) ( x − 3 )$]
+ Find two numbers whose product is #math.equation(block: false, alt: "minus 12")[$− 12$] and whose sum is #math.equation(block: false, alt: "minus 1")[$− 1$]. Because the product is negative, the two numbers must be of opposite sign and their sum must be #math.equation(block: false, alt: "minus 1")[$− 1$]. By listing the possible factors of #math.equation(block: false, alt: "minus 12")[$− 12$], we find that the two numbers are #math.equation(block: false, alt: "minus 4")[$− 4$] and #math.equation(block: false, alt: "3")[$3$]. Hence,
                #math.equation(block: true, alt: "x squared minus x minus 12 equals open parenthesis x minus 4 close parenthesis open parenthesis x plus 3 close parenthesis")[$x^(2) − x − 12 = ( x − 4 ) ( x + 3 )$]
]
]


    If the coefficient of the quadratic term is not #math.equation(block: false, alt: "1")[$1$], we must also consider its factors.


    #examplebox("Example 2")[][
Factor #math.equation(block: false, alt: "8 x squared minus 9 minus 21 x")[$" " 8 x^(2) − 9 − 21 x$]


      
      #solutionbox[
+ Write the trinomial in decreasing powers of #math.equation(block: false, alt: "x")[$x$].
                #math.equation(block: true, alt: "8 x squared minus 21 x minus 9")[$8 x^(2) − 21 x − 9$]
+ List the possible factors for the quadratic term.
                #math.equation(block: true, alt: "open parenthesis 8 x close parenthesis open parenthesis x close parenthesis; open parenthesis 4 x close parenthesis open parenthesis 2 x close parenthesis")[$( 8 x #hide($00000$) ) ( x #hide($000000$) ) \ ( 4 x #hide($00000$) ) ( 2 x #hide($00000$) )$]
+ Consider possible factors for the constant term: #math.equation(block: false, alt: "9")[$9$] may be factored as #math.equation(block: false, alt: "9 times 1")[$9 ⋅ 1$] or as #math.equation(block: false, alt: "3 times 3")[$3 ⋅ 3$]. Form all possible pairs of binomial factor using these factorizations.#figure(figph[list of possible binomial factor pairs], alt: "list of possible binomial factor pairs", caption: none)
+ Select the combinations of the products  #math.equation(block: false, alt: "①")[$①$] and #math.equation(block: false, alt: "②")[$②$] whose sum or difference could be the linear term, #math.equation(block: false, alt: "minus 21 x")[$− 21 x$].
                #math.equation(block: true, alt: "open parenthesis 8 x 3 close parenthesis open parenthesis x 3 close parenthesis")[$( 8 x #hide($000$) 3 ) ( x #hide($000$) 3 )$]
+ Insert the proper signs:
                #math.equation(block: true, alt: "open parenthesis 8 x plus 3 close parenthesis open parenthesis x minus 3 close parenthesis")[$( 8 x + 3 ) ( x − 3 )$]
]
]


    With practice, you can usually factor trinomials of the form #math.equation(block: false, alt: "A x squared plus B x plus C")[$A x^(2) + B x + C$] mentally. The following observations may help.


    #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
+ If #math.equation(block: false, alt: "A")[$A$], #math.equation(block: false, alt: "B")[$B$] and #math.equation(block: false, alt: "C")[$C$] are all positive, both signs in the factored form are positive. For example, as a first step in factoring #math.equation(block: false, alt: "6 x squared plus 11 x plus 4")[$6 x^(2) + 11 x + 4$], we could write
            #math.equation(block: true, alt: "open parenthesis plus close parenthesis open parenthesis plus close parenthesis")[$( #hide($000$) + #hide($000$) ) ( #hide($000$) + #hide($000$) )$]
+ If #math.equation(block: false, alt: "A")[$A$] and #math.equation(block: false, alt: "C")[$C$] are positive and #math.equation(block: false, alt: "B")[$B$] is negative, both signs in the factored form are negative. Thus as a first step in factoring #math.equation(block: false, alt: "6 x squared minus 11 x plus 4")[$6 x^(2) − 11 x + 4$], we could write
            #math.equation(block: true, alt: "open parenthesis minus close parenthesis open parenthesis minus close parenthesis")[$( #hide($000$) − #hide($000$) ) ( #hide($000$) − #hide($000$) )$]
+ If #math.equation(block: false, alt: "C")[$C$] is negative, the signs in the factored form are opposite. Thus as a first step in factoring #math.equation(block: false, alt: "6 x squared minus 5 x minus 4")[$6 x^(2) − 5 x − 4$], we could write
            #math.equation(block: true, alt: "open parenthesis plus close parenthesis open parenthesis minus close parenthesis or open parenthesis minus close parenthesis open parenthesis plus close parenthesis")[$( #hide($000$) + #hide($000$) ) ( #hide($000$) − #hide($000$) ) " " " " " or " " " " " ( #hide($000$) − #hide($000$) ) ( #hide($000$) + #hide($000$) )$]
]


    #examplebox("Example 3")[][
+ #math.equation(block: false, alt: "6 x squared plus 5 x plus 1, equals open parenthesis plus close parenthesis open parenthesis plus close parenthesis; equals open parenthesis 3 x plus 1 close parenthesis open parenthesis 2 x plus 1 close parenthesis")[$6 x^(2) + 5 x + 1 & = ( #hide($00$) + #hide($00$) ) ( #hide($00$) + #hide($00$) ) \  & = ( 3 x + 1 ) ( 2 x + 1 )$]
+ #math.equation(block: false, alt: "6 x squared minus 5 x plus 1, equals open parenthesis minus close parenthesis open parenthesis minus close parenthesis; equals open parenthesis 3 x minus 1 close parenthesis open parenthesis 2 x minus 1 close parenthesis")[$6 x^(2) − 5 x + 1 & = ( #hide($00$) − #hide($00$) ) ( #hide($00$) − #hide($00$) ) \  & = ( 3 x − 1 ) ( 2 x − 1 )$]
+ #math.equation(block: false, alt: "6 x squared minus x minus 1, equals open parenthesis plus close parenthesis open parenthesis minus close parenthesis; equals open parenthesis 3 x plus 1 close parenthesis open parenthesis 2 x minus 1 close parenthesis")[$6 x^(2) − x − 1 & = ( #hide($00$) + #hide($00$) ) ( #hide($00$) − #hide($00$) ) \  & = ( 3 x + 1 ) ( 2 x − 1 )$]
+ #math.equation(block: false, alt: "6 x squared minus x y minus y squared, equals open parenthesis plus close parenthesis open parenthesis minus close parenthesis; equals open parenthesis 3 x plus y close parenthesis open parenthesis 2 x minus y close parenthesis")[$6 x^(2) − x y − y^(2) & = ( #hide($00$) + #hide($00$) ) ( #hide($00$) − #hide($00$) ) \  & = ( 3 x + y ) ( 2 x − y )$]
]


    
      
=== Special Products and Factors


      The products below are special cases of the multiplication of binomials. They occur so often that you should learn to recognize them on sight.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Special Products]


        + #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared equals open parenthesis a plus b close parenthesis open parenthesis a plus b close parenthesis equals a squared plus 2 a b plus b squared")[$( a + b )^(2) = ( a + b ) ( a + b ) = a^(2) + 2 a b + b^(2)$]
+ #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis squared equals open parenthesis a minus b close parenthesis open parenthesis a minus b close parenthesis equals a squared minus 2 a b plus b squared")[$( a − b )^(2) = ( a − b ) ( a − b ) = a^(2) − 2 a b + b^(2)$]
+ #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis open parenthesis a minus b close parenthesis equals a squared minus b squared")[$( a + b ) ( a − b ) = a^(2) − b^(2)$]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
Notice that in (I) #math.equation(block: false, alt: "open parenthesis a plus b close parenthesis squared not equal to a squared plus b squared")[$" " " " ( a + b )^(2) ≠ a^(2) + b^(2) " " " "$], and that in (II) #math.equation(block: false, alt: "open parenthesis a minus b close parenthesis squared not equal to a squared minus b squared")[$" " " " ( a − b )^(2) ≠ a^(2) − b^(2) " " " "$]. For example,


        #math.equation(block: true, alt: "open parenthesis x plus 4 close parenthesis squared, not equal to x squared plus 16 ,, instead, open parenthesis x plus 4 close parenthesis squared, equals x squared plus 8 x plus 16; open parenthesis t minus 5 close parenthesis squared, not equal to t squared minus 16 ,, instead, open parenthesis t minus 5 close parenthesis squared, equals t squared minus 10 t plus 25")[$( x + 4 )^(2) & ≠ x^(2) + 16 , &  & " instead " & ( x + 4 )^(2) & = x^(2) + 8 x + 16 \ ( t − 5 )^(2) & ≠ t^(2) − 16 , &  & " instead " & ( t − 5 )^(2) & = t^(2) − 10 t + 25$]
]


      #examplebox("Example 4")[][
+ #math.equation(block: false, alt: "3 open parenthesis x plus 4 close parenthesis squared, equals 3 open parenthesis x squared plus 2 times 4 x plus 4 squared close parenthesis; equals 3 x squared plus 24 x plus 48")[$3 ( x + 4 )^(2) & = 3 ( x^(2) + 2 ⋅ 4 x + 4^(2) ) \  & = 3 x^(2) + 24 x + 48$]
+ #math.equation(block: false, alt: "open parenthesis y plus 5 close parenthesis open parenthesis y minus 5 close parenthesis, equals y squared minus 5 squared; equals y squared minus 25")[$( y + 5 ) ( y − 5 ) & = y^(2) − 5^(2) \  & = y^(2) − 25$]
+ #math.equation(block: false, alt: "open parenthesis 3 x minus 2 y close parenthesis squared, equals open parenthesis 3 x close parenthesis squared minus 2 open parenthesis 3 x close parenthesis open parenthesis 2 y close parenthesis plus open parenthesis 2 y close parenthesis squared; equals 9 x squared minus 12 x y plus 4 y squared")[$( 3 x − 2 y )^(2) & = ( 3 x )^(2) − 2 ( 3 x ) ( 2 y ) + ( 2 y )^(2) \  & = 9 x^(2) − 12 x y + 4 y^(2)$]
]


      Each of the formulas for special products, when viewed from right to left, also represents a special case of factoring quadratic polynomials.


      #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#emph[Special Factorizations]


        + #math.equation(block: false, alt: "a squared plus 2 a b plus b squared equals open parenthesis a plus b close parenthesis squared")[$a^(2) + 2 a b + b^(2) = ( a + b )^(2)$]
+ #math.equation(block: false, alt: "a squared minus 2 a b plus b squared equals open parenthesis a minus b close parenthesis squared")[$a^(2) − 2 a b + b^(2) = ( a − b )^(2)$]
+ #math.equation(block: false, alt: "a squared minus b squared equals open parenthesis a plus b close parenthesis open parenthesis a minus b close parenthesis")[$a^(2) − b^(2) = ( a + b ) ( a − b )$]
+ #math.equation(block: false, alt: "a squared plus b squared cannot be factored")[$a^(2) + b^(2) " " " " " cannot be factored"$]
]


      The trinomials in (I) and (II) are sometimes called #strong[perfect-square trinomials] because they are squares of binomials. Note that the sum of two squares, #math.equation(block: false, alt: "a squared plus b squared")[$a^(2) + b^(2)$], cannot be factored.


      #examplebox("Example 5")[][
Factor.


          + #math.equation(block: false, alt: "x squared plus 8 x plus 16")[$x^(2) + 8 x + 16$]
+ #math.equation(block: false, alt: "y squared minus 10 y plus 25")[$y^(2) − 10 y + 25$]
+ #math.equation(block: false, alt: "4 a squared minus 12 a b plus 9 b squared")[$4 a^(2) − 12 a b + 9 b^(2)$]
+ #math.equation(block: false, alt: "25 m squared n squared plus 20 m n plus 4")[$25 m^(2) n^(2) + 20 m n + 4$]


        
        #solutionbox[
+ Because #math.equation(block: false, alt: "16")[$16$] is equal to #math.equation(block: false, alt: "4 squared")[$4^(2)$] and #math.equation(block: false, alt: "8")[$8$] is equal to #math.equation(block: false, alt: "2 times 4")[$2 ⋅ 4$],
                #math.equation(block: true, alt: "x squared plus 8 x plus 16, equals x squared minus 2 times 4 x plus 4 squared; equals open parenthesis x plus 4 close parenthesis squared")[$x^(2) + 8 x + 16 & = x^(2) − 2 ⋅ 4 x + 4^(2) \  & = ( x + 4 )^(2)$]
+ Because #math.equation(block: false, alt: "25 equals 5 squared")[$25 = 5^(2)$]  and #math.equation(block: false, alt: "10 equals 2 times 5")[$10 = 2 ⋅ 5$],
                #math.equation(block: true, alt: "y squared minus 10 y plus 25, equals y squared minus 2 times 5 y plus 5 squared; equals open parenthesis y minus 5 close parenthesis squared")[$y^(2) − 10 y + 25 & = y^(2) − 2 ⋅ 5 y + 5^(2) \  & = ( y − 5 )^(2)$]
+ Because #math.equation(block: false, alt: "4 a squared equals open parenthesis 2 a close parenthesis squared")[$4 a^(2) = ( 2 a )^(2)$], #math.equation(block: false, alt: "9 b squared equals open parenthesis 3 b close parenthesis squared")[$9 b^(2) = ( 3 b )^(2)$],  and #math.equation(block: false, alt: "2 a b equals 2 open parenthesis 2 a close parenthesis open parenthesis 3 b close parenthesis")[$2 a b = 2 ( 2 a ) ( 3 b )$],
                #math.equation(block: true, alt: "4 a squared minus 12 a b plus 9 b squared, equals open parenthesis 2 a close parenthesis squared minus 2 open parenthesis 2 a close parenthesis open parenthesis 3 b close parenthesis plus open parenthesis 3 b close parenthesis squared; equals open parenthesis 2 a minus 3 b close parenthesis squared")[$4 a^(2) − 12 a b + 9 b^(2) & = ( 2 a )^(2) − 2 ( 2 a ) ( 3 b ) + ( 3 b )^(2) \  & = ( 2 a − 3 b )^(2)$]
+ Because #math.equation(block: false, alt: "25 m squared n squared equals open parenthesis 5 m n close parenthesis squared")[$25 m^(2) n^(2) = ( 5 m n )^(2)$], #math.equation(block: false, alt: "4 equals 2 squared")[$4 = 2^(2)$],  and #math.equation(block: false, alt: "20 m n equals 2 open parenthesis 5 m n close parenthesis open parenthesis 2 close parenthesis")[$20 m n = 2 ( 5 m n ) ( 2 )$],
                #math.equation(block: true, alt: "25 m squared n squared plus 20 m n plus 4, equals open parenthesis 5 m n close parenthesis squared plus 2 open parenthesis 5 m n close parenthesis open parenthesis 2 close parenthesis plus 2 squared; equals open parenthesis 5 m n plus 2 close parenthesis squared")[$25 m^(2) n^(2) + 20 m n + 4 & = ( 5 m n )^(2) + 2 ( 5 m n ) ( 2 ) + 2^(2) \  & = ( 5 m n + 2 )^(2)$]
]
]


      Binomials of the form #math.equation(block: false, alt: "a squared minus b squared")[$a^(2) − b^(2)$] are often called the #strong[difference of two squares].


      #examplebox("Example 6")[][
Factor if possible.


          + #math.equation(block: false, alt: "x squared minus 81")[$x^(2) − 81$]
+ #math.equation(block: false, alt: "4 x squared minus 9 y squared")[$4 x^(2) − 9 y^(2)$]
+ #math.equation(block: false, alt: "x squared plus 81")[$x^(2) + 81$]


        
        #solutionbox[
+ The expression #math.equation(block: false, alt: "x squared minus 81")[$x^(2) − 81$] is the difference of two squares, #math.equation(block: false, alt: "x squared minus 9 squared")[$x^(2) − 9^(2)$], and thus can be factored according to Special Factorization (III) above.
                #math.equation(block: true, alt: "x squared minus 81, equals x squared minus 9 squared; equals open parenthesis x plus 9 close parenthesis open parenthesis x minus 9 close parenthesis")[$x^(2) − 81 & = x^(2) − 9^(2) \  & = ( x + 9 ) ( x − 9 )$]
+ Because #math.equation(block: false, alt: "4 x squared minus 9 y squared")[$4 x^(2) − 9 y^(2)$] can be written as #math.equation(block: false, alt: "open parenthesis 2 x close parenthesis squared minus open parenthesis 3 y close parenthesis squared")[$( 2 x )^(2) − ( 3 y )^(2)$],
                #math.equation(block: true, alt: "4 x squared minus 9 y squared, equals open parenthesis 2 x squared close parenthesis minus open parenthesis 3 y close parenthesis squared; equals open parenthesis 2 x plus 3 y close parenthesis open parenthesis 2 x minus 3 y close parenthesis")[$4 x^(2) − 9 y^(2) & = ( 2 x^(2) ) − ( 3 y )^(2) \  & = ( 2 x + 3 y ) ( 2 x − 3 y )$]
+ The expression #math.equation(block: false, alt: "x squared plus 81")[$x^(2) + 81$], or #math.equation(block: false, alt: "x squared plus 0 x plus 81")[$x^(2) + 0 x + 81$], is #emph[not] factorable, because no two real numbers have a product of #math.equation(block: false, alt: "81")[$81$] and a sum of #math.equation(block: false, alt: "0")[$0$].
]
]


      #notebox("Caution", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[
#math.equation(block: false, alt: "x squared plus 81 not equal to open parenthesis x plus 9 close parenthesis open parenthesis x plus 9 close parenthesis")[$x^(2) + 81 ≠ ( x + 9 ) ( x + 9 )$], which you can verify by multiplying


        #math.equation(block: true, alt: "open parenthesis x plus 9 close parenthesis open parenthesis x plus 9 close parenthesis equals x squared plus 18 x plus 8")[$( x + 9 ) ( x + 9 ) = x^(2) + 18 x + 8$]
]


      The factors #math.equation(block: false, alt: "x plus 9")[$x + 9$] and #math.equation(block: false, alt: "x minus 9")[$x − 9$] in Examplea are called #strong[conjugates] of each other. In general, any binomials of the form #math.equation(block: false, alt: "a plus b")[$a + b$] and #math.equation(block: false, alt: "a minus b")[$a − b$] are called a #strong[conjugate pair].


    
    
      
=== Section Summary


      
        
==== Vocabulary


        Look up the definitions of new terms in the Glossary.


        - Perfect-square trinomial
- Difference of squares
- Conjugate


      
      
        
==== SKILLS


        Practice each skill in the exercises listed.


        + Factor quadratic trinomials: \#1–36
+ Expand special products: \#37–48
+ Factor special quadratic expressions: \#49–68


      
    
    
      
=== Exercises A.8


      For Problems 1-36, factor completely.


      
        #math.equation(block: true, alt: "x squared plus 5 x plus 6")[$x^(2) + 5 x + 6$]


        
          #math.equation(block: true, alt: "open parenthesis x plus 2 close parenthesis open parenthesis x plus 3 close parenthesis")[$( x + 2 ) ( x + 3 )$]


        
      
      
        #math.equation(block: true, alt: "x squared plus 5 x plus 4")[$x^(2) + 5 x + 4$]


      
      
        #math.equation(block: true, alt: "y squared minus 7 y plus 12")[$y^(2) − 7 y + 12$]


        
          #math.equation(block: true, alt: "open parenthesis y minus 3 close parenthesis open parenthesis y minus 4 close parenthesis")[$( y − 3 ) ( y − 4 )$]


        
      
      
        #math.equation(block: true, alt: "y squared minus 7 y plus 10")[$y^(2) − 7 y + 10$]


      
      
        #math.equation(block: true, alt: "x squared minus 6 minus x")[$x^(2) − 6 − x$]


        
          #math.equation(block: true, alt: "open parenthesis x minus 3 close parenthesis open parenthesis x plus 2 close parenthesis")[$( x − 3 ) ( x + 2 )$]


        
      
      
        #math.equation(block: true, alt: "x squared minus 15 minus 2 x")[$x^(2) − 15 − 2 x$]


      
      
        #math.equation(block: true, alt: "2 x squared plus 3 x minus 2")[$2 x^(2) + 3 x − 2$]


        
          #math.equation(block: true, alt: "open parenthesis 2 x minus 1 close parenthesis open parenthesis x plus 2 close parenthesis")[$( 2 x − 1 ) ( x + 2 )$]


        
      
      
        #math.equation(block: true, alt: "3 x squared minus 7 x plus 2")[$3 x^(2) − 7 x + 2$]


      
      
        #math.equation(block: true, alt: "7 x plus 4 x squared minus 2")[$7 x + 4 x^(2) − 2$]


        
          #math.equation(block: true, alt: "open parenthesis 4 x minus 1 close parenthesis open parenthesis x plus 2 close parenthesis")[$( 4 x − 1 ) ( x + 2 )$]


        
      
      
        #math.equation(block: true, alt: "1 minus 5 x plus 6 x squared")[$1 − 5 x + 6 x^(2)$]


      
      
        #math.equation(block: true, alt: "9 y squared minus 21 y minus 8")[$9 y^(2) − 21 y − 8$]


        
          #math.equation(block: true, alt: "open parenthesis 3 y plus 1 close parenthesis open parenthesis 3 y minus 8 close parenthesis")[$( 3 y + 1 ) ( 3 y − 8 )$]


        
      
      
        #math.equation(block: true, alt: "10 y squared minus 3 y minus 18")[$10 y^(2) − 3 y − 18$]


      
      
        #math.equation(block: true, alt: "10 u squared minus 3 minus u")[$10 u^(2) − 3 − u$]


        
          #math.equation(block: true, alt: "open parenthesis 2 u plus 1 close parenthesis open parenthesis 5 u minus 3 close parenthesis")[$( 2 u + 1 ) ( 5 u − 3 )$]


        
      
      
        #math.equation(block: true, alt: "8 u squared minus 3 plus 5 u")[$8 u^(2) − 3 + 5 u$]


      
      
        #math.equation(block: true, alt: "21 x squared minus 43 x minus 14")[$21 x^(2) − 43 x − 14$]


        
          #math.equation(block: true, alt: "open parenthesis 3 x minus 7 close parenthesis open parenthesis 7 x plus 2 close parenthesis")[$( 3 x − 7 ) ( 7 x + 2 )$]


        
      
      
        #math.equation(block: true, alt: "24 x squared minus 29 x plus 5")[$24 x^(2) − 29 x + 5$]


      
      
        #math.equation(block: true, alt: "5 a plus 72 a squared minus 25")[$5 a + 72 a^(2) − 25$]


        
          #math.equation(block: true, alt: "open parenthesis 9 a plus 4 close parenthesis open parenthesis 8 a minus 3 close parenthesis")[$( 9 a + 4 ) ( 8 a − 3 )$]


        
      
      
        #math.equation(block: true, alt: "minus 30 a plus 72 a squared minus 25")[$− 30 a + 72 a^(2) − 25$]


      
      
        #math.equation(block: true, alt: "12 minus 53 x plus 30 x squared")[$12 − 53 x + 30 x^(2)$]


        
          #math.equation(block: true, alt: "open parenthesis 2 x minus 3 close parenthesis open parenthesis 15 x minus 4 close parenthesis")[$( 2 x − 3 ) ( 15 x − 4 )$]


        
      
      
        #math.equation(block: true, alt: "39 x plus 80 x squared minus 20")[$39 x + 80 x^(2) − 20$]


      
      
        #math.equation(block: true, alt: "minus 30 t minus 44 plus 54 t squared")[$− 30 t − 44 + 54 t^(2)$]


        
          #math.equation(block: true, alt: "2 open parenthesis 3 t plus 2 close parenthesis open parenthesis 9 t minus 11 close parenthesis")[$2 ( 3 t + 2 ) ( 9 t − 11 )$]


        
      
      
        #math.equation(block: true, alt: "48 t squared minus 122 t plus 39")[$48 t^(2) − 122 t + 39$]


      
      
        #math.equation(block: true, alt: "3 x squared minus 7 a x plus 2 a squared")[$3 x^(2) − 7 a x + 2 a^(2)$]


        
          #math.equation(block: true, alt: "open parenthesis x minus 2 a close parenthesis open parenthesis 3 x minus a close parenthesis")[$( x − 2 a ) ( 3 x − a )$]


        
      
      
        #math.equation(block: true, alt: "9 x squared plus 9 a x minus 10 a squared")[$9 x^(2) + 9 a x − 10 a^(2)$]


      
      
        #math.equation(block: true, alt: "15 x squared minus 4 x y minus 4 y squared")[$15 x^(2) − 4 x y − 4 y^(2)$]


        
          #math.equation(block: true, alt: "open parenthesis 3 x minus 2 y close parenthesis open parenthesis 5 x plus 2 y close parenthesis")[$( 3 x − 2 y ) ( 5 x + 2 y )$]


        
      
      
        #math.equation(block: true, alt: "12 x squared plus 7 x y minus 12 y squared")[$12 x^(2) + 7 x y − 12 y^(2)$]


      
      
        #math.equation(block: true, alt: "18 u squared plus 20 v squared minus 39 u v")[$18 u^(2) + 20 v^(2) − 39 u v$]


        
          #math.equation(block: true, alt: "open parenthesis 3 u minus 4 v close parenthesis open parenthesis 6 u minus 5 v close parenthesis")[$( 3 u − 4 v ) ( 6 u − 5 v )$]


        
      
      
        #math.equation(block: true, alt: "24 u squared minus 20 v squared plus 17 u v")[$24 u^(2) − 20 v^(2) + 17 u v$]


      
      
        #math.equation(block: true, alt: "12 a squared minus 14 b squared minus 13 a b")[$12 a^(2) − 14 b^(2) − 13 a b$]


        
          #math.equation(block: true, alt: "open parenthesis 3 a plus 2 b close parenthesis open parenthesis 4 a minus 7 b close parenthesis")[$( 3 a + 2 b ) ( 4 a − 7 b )$]


        
      
      
        #math.equation(block: true, alt: "24 a squared minus 15 b squared minus 2 a b")[$24 a^(2) − 15 b^(2) − 2 a b$]


      
      
        #math.equation(block: true, alt: "10 a squared b squared minus 19 a b plus 6")[$10 a^(2) b^(2) − 19 a b + 6$]


        
          #math.equation(block: true, alt: "open parenthesis 5 a b minus 2 close parenthesis open parenthesis 2 a b minus 3 close parenthesis")[$( 5 a b − 2 ) ( 2 a b − 3 )$]


        
      
      
        #math.equation(block: true, alt: "12 a squared b squared minus a b minus 20")[$12 a^(2) b^(2) − a b − 20$]


      
      
        #math.equation(block: true, alt: "56 x squared y squared minus 2 x y minus 4")[$56 x^(2) y^(2) − 2 x y − 4$]


        
          #math.equation(block: true, alt: "2 open parenthesis 4 x y plus 1 close parenthesis open parenthesis 7 x y minus 2 close parenthesis")[$2 ( 4 x y + 1 ) ( 7 x y − 2 )$]


        
      
      
        #math.equation(block: true, alt: "54 x squared y squared plus 3 x y minus 2")[$54 x^(2) y^(2) + 3 x y − 2$]


      
      
        #math.equation(block: true, alt: "22 a squared z squared minus 21 minus 19 a z")[$22 a^(2) z^(2) − 21 − 19 a z$]


        
          #math.equation(block: true, alt: "open parenthesis 2 a z minus 3 close parenthesis open parenthesis 11 a z plus 7 close parenthesis")[$( 2 a z − 3 ) ( 11 a z + 7 )$]


        
      
      
        #math.equation(block: true, alt: "26 a squared z squared minus 24 plus 23 a z")[$26 a^(2) z^(2) − 24 + 23 a z$]


      
      For Problems 37-48, write the expression as a polynomial and simplify.


      
        #math.equation(block: true, alt: "open parenthesis x plus 3 close parenthesis squared")[$( x + 3 )^(2)$]


        
          #math.equation(block: true, alt: "x squared plus 6 x plus 9")[$x^(2) + 6 x + 9$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis y minus 4 close parenthesis squared")[$( y − 4 )^(2)$]


      
      
        #math.equation(block: true, alt: "open parenthesis 2 y minus 5 close parenthesis squared")[$( 2 y − 5 )^(2)$]


        
          #math.equation(block: true, alt: "4 y squared minus 20 y plus 25")[$4 y^(2) − 20 y + 25$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 3 x plus 2 close parenthesis squared")[$( 3 x + 2 )^(2)$]


      
      
        #math.equation(block: true, alt: "open parenthesis x plus 3 close parenthesis open parenthesis x minus 3 close parenthesis")[$( x + 3 ) ( x − 3 )$]


        
          #math.equation(block: true, alt: "x squared minus 9")[$x^(2) − 9$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis x minus 7 close parenthesis open parenthesis x plus 7 close parenthesis")[$( x − 7 ) ( x + 7 )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 3 t minus 4 s close parenthesis open parenthesis 3 t plus 4 s close parenthesis")[$( 3 t − 4 s ) ( 3 t + 4 s )$]


        
          #math.equation(block: true, alt: "9 t squared minus 16 s squared")[$9 t^(2) − 16 s^(2)$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 2 x plus a close parenthesis open parenthesis 2 x minus a close parenthesis")[$( 2 x + a ) ( 2 x − a )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 5 a minus 2 close parenthesis open parenthesis 5 a minus 2 close parenthesis")[$( 5 a − 2 ) ( 5 a − 2 )$]


        
          #math.equation(block: true, alt: "25 a squared minus 20 a b plus 4 b squared")[$25 a^(2) − 20 a b + 4 b^(2)$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 4 u plus 5 v close parenthesis open parenthesis 4 u plus 5 v close parenthesis")[$( 4 u + 5 v ) ( 4 u + 5 v )$]


      
      
        #math.equation(block: true, alt: "open parenthesis 8 x z plus 3 close parenthesis open parenthesis 8 x z plus 3 close parenthesis")[$( 8 x z + 3 ) ( 8 x z + 3 )$]


        
          #math.equation(block: true, alt: "64 x squared z squared plus 48 x z plus 9")[$64 x^(2) z^(2) + 48 x z + 9$]


        
      
      
        #math.equation(block: true, alt: "open parenthesis 7 y z minus 2 close parenthesis open parenthesis 7 y z minus 2 close parenthesis")[$( 7 y z − 2 ) ( 7 y z − 2 )$]


      
      For Problems 49-68, factor completely.


      
        #math.equation(block: true, alt: "x squared minus 25")[$x^(2) − 25$]


        
          #math.equation(block: true, alt: "open parenthesis x plus 5 close parenthesis open parenthesis x minus 5 close parenthesis")[$( x + 5 ) ( x − 5 )$]


        
      
      
        #math.equation(block: true, alt: "x squared minus 36")[$x^(2) − 36$]


      
      
        #math.equation(block: true, alt: "x squared minus 24 x plus 144")[$x^(2) − 24 x + 144$]


        
          #math.equation(block: true, alt: "open parenthesis x minus 12 close parenthesis squared")[$( x − 12 )^(2)$]


        
      
      
        #math.equation(block: true, alt: "x squared plus 26 x plus 169")[$x^(2) + 26 x + 169$]


      
      
        #math.equation(block: true, alt: "x squared minus 4 y squared")[$x^(2) − 4 y^(2)$]


        
          #math.equation(block: true, alt: "open parenthesis x plus 2 y close parenthesis open parenthesis x minus 2 y close parenthesis")[$( x + 2 y ) ( x − 2 y )$]


        
      
      
        #math.equation(block: true, alt: "9 x squared minus y squared")[$9 x^(2) − y^(2)$]


      
      
        #math.equation(block: true, alt: "4 x squared plus 12 x plus 9")[$4 x^(2) + 12 x + 9$]


        
          #math.equation(block: true, alt: "open parenthesis 2 x plus 3 close parenthesis squared")[$( 2 x + 3 )^(2)$]


        
      
      
        #math.equation(block: true, alt: "4 y squared plus 4 y plus 1")[$4 y^(2) + 4 y + 1$]


      
      
        #math.equation(block: true, alt: "9 u squared minus 30 u v plus 25 v squared")[$9 u^(2) − 30 u v + 25 v^(2)$]


        
          #math.equation(block: true, alt: "open parenthesis 3 u minus 5 v close parenthesis squared")[$( 3 u − 5 v )^(2)$]


        
      
      
        #math.equation(block: true, alt: "16 s squared minus 56 s t plus 49 t squared")[$16 s^(2) − 56 s t + 49 t^(2)$]


      
      
        #math.equation(block: true, alt: "4 a squared minus 25 b squared")[$4 a^(2) − 25 b^(2)$]


        
          #math.equation(block: true, alt: "open parenthesis 2 a plus 5 b close parenthesis open parenthesis 2 a minus 5 b close parenthesis")[$( 2 a + 5 b ) ( 2 a − 5 b )$]


        
      
      
        #math.equation(block: true, alt: "16 a squared minus 9 b squared")[$16 a^(2) − 9 b^(2)$]


      
      
        #math.equation(block: true, alt: "x squared y squared minus 81")[$x^(2) y^(2) − 81$]


        
          #math.equation(block: true, alt: "open parenthesis x y plus 9 close parenthesis open parenthesis x y minus 9 close parenthesis")[$( x y + 9 ) ( x y − 9 )$]


        
      
      
        #math.equation(block: true, alt: "x squared y squared minus 64")[$x^(2) y^(2) − 64$]


      
      
        #math.equation(block: true, alt: "9 x squared y squared plus 6 x y plus 1")[$9 x^(2) y^(2) + 6 x y + 1$]


        
          #math.equation(block: true, alt: "open parenthesis 3 x y plus 1 close parenthesis squared")[$( 3 x y + 1 )^(2)$]


        
      
      
        #math.equation(block: true, alt: "4 x squared y squared plus 12 x y plus 9")[$4 x^(2) y^(2) + 12 x y + 9$]


      
      
        #math.equation(block: true, alt: "16 squared y squared minus 1")[$16^(2) y^(2) − 1$]


        
          #math.equation(block: true, alt: "open parenthesis 4 x y minus 1 close parenthesis open parenthesis 4 x y plus 1 close parenthesis")[$( 4 x y − 1 ) ( 4 x y + 1 )$]


        
      
      
        #math.equation(block: true, alt: "64 x squared y squared minus 1")[$64 x^(2) y^(2) − 1$]


      
      
        #math.equation(block: true, alt: "open parenthesis x plus 2 close parenthesis squared minus y squared")[$( x + 2 )^(2) − y^(2)$]


        
          #math.equation(block: true, alt: "open parenthesis x plus 2 minus y close parenthesis open parenthesis x plus 2 plus y close parenthesis")[$( x + 2 − y ) ( x + 2 + y )$]


        
      
      
        #math.equation(block: true, alt: "x squared minus open parenthesis y minus 3 close parenthesis squared")[$x^(2) − ( y − 3 )^(2)$]