Factoring Binomials by coltonvelencia

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									DEV108 Introduction to Algebra                       Unit 4: Exponents, Variables, and Polynomials

     Lesson 11: Greatest Common Monomial Factors, Multiplying Binomials,
     Factoring Trinomials, Rational Expressions
Instructions:
       Follow along with video and fill in the blanks as indicated. Space has been provided for
       you to show all work on this sheet and take any additional notes.

Greatest Common Monomial Factors

Examples: Copy steps/answers from video.
                                                             This example also has common variables
          2
1.    24m + 12m – 8                                          2.    24m3 + 12m2 – 8m




                 Fill in the blank(s):

              Factoring Out the Greatest Common Monomial Factor
                         1. Find the largest numerical factor that can be divided into each term.
                           2. Determine the largest common variable factor by choosing the
                     ________exponent on any variable that occurs in all terms.
                           3. Then _____________ each term in the expression by the product of the
                     factors obtained in steps 1 and 2. The remaining factors form a polynomial that will
                     be enclosed in ________________________.


Copy steps/answers from video.

3.    25m5 – 15m4 + 18m3                                     4.    12x2y + 16y2 – 4y




5.    14a2b – 7ab2 + 28ab




6.    5x(3x2 – 2x + 1)                                       7.    6a2b(4ab2 – 2a2 + 3)




                                                                                                            1
                  STOP THE LESSON AND WORK THE PROBLEM SET

                           Problem Set: Greatest Common Monomial Factors
Factor out the greatest common monomial factor.

1.       6m2 + 4mn + 2m3                            2.      27a3 – 18a2                                 3.      30a3b2 – 18a2b2 – 6ab




Multiply.

4.       4x2(5x3 + 2x – 7)                          5.      3p(p2 + 7pq – q2)                           6.      5x2y2(2xy – 3)




             RESUME THE LESSON FOR ANSWERS AND SOLUTIONS
     (Note: On the video you will first see the answers only. Following the answers, solution steps for all problems are also shown on the video.)


Lesson 11 (cont’d): Multiplying Binomials
Examples: Copy steps/answers from video.

1.       (x + 5)(x + 7)                                                            2.      (y – 3)(y – 4)




                                     Fill in the blank(s):

                                          F. O. I. L. – For Multiplying Binomials
                                              F – multiply the F_________ terms
                                             O – multiply the O_________ terms
                                              I – multiply the I__________ terms
                                              L – Multiply the L_________ terms


Examples: Copy steps/answers from video.

3.       (x + 8)(x + 6)                             4.      (y – 12)(y – 4)                             5.      (p – 8)(p + 9)




                                                                                                                                                     2
                  STOP THE LESSON AND WORK THE PROBLEM SET
                                         Problem Set: Multiplying Binomials
Multiply using the vertical method.
1.       (m – 4)(m + 9)                                                            2.      (r + 1)(r + 3)




Multiply using the FOIL method.
3.       (p – 5)(p – 12)                            4.      (s – 3)(s + 9)                              5.      (y + 5)(y + 8)




             RESUME THE LESSON FOR ANSWERS AND SOLUTIONS
     (Note: On the video you will first see the answers only. Following the answers, solution steps for all problems are also shown on the video.)




Lesson 11 (cont’d): Factoring Trinomials
Examples: Copy steps/answers from video.
1.       x2 + 10x + 24                              2.      x2 + 9x + 8                                 3.      x2 – 10x + 21




4.       x2 – 5x + 4                                5.      x2 – 4x – 21                                6.      x2 + 3x – 4




                     Fill in the blank(s):

                  Sign Clues for Factoring Trinomials of the Form x2 + bx + c
                                When the last term is positive . . .
                                  The factors will have the___________ sign and it will ______________ the sign of
                           the ____________ term.
                                 When the last term is negative . . .
                                  The factors will have ________________signs and the signs will be placed so that
                           when combined, they equal the________________ of the _____________________
                           term.                                                                               3
Copy steps/answers from video.

7.       x2 – 2x – 80                                                              8.      x2 – 15x + 54




9.       2x2 + 4x – 30                                                             10.     3x3 + 36x2 + 105x




                        Fill in the blank(s):

                       Steps in Factoring:
                                    1. Factor out the greatest _____________ factor, if any.
                                    2. If the remaining expression is a trinomial, look for possible
                                _____________ _______________.
                                    3. Polynomials that are not factorable are called _____________.



                  STOP THE LESSON AND WORK THE PROBLEM SET

                                          Problem Set: Factoring Trinomials
Factor completely.

1.       x2 – 14x + 49                                                             2.      x2 + 4x – 96




3.       p2 + 14p + 48                                                             4.      4x3 – 32x2 + 28x




5.       y2 – 9y – 36                                                              6.      2q2 + 6q – 56




             RESUME THE LESSON FOR ANSWERS AND SOLUTIONS
     (Note: On the video you will first see the answers only. Following the answers, solution steps for all problems are also shown on the video.)



                                                                                                                                                     4
Lesson 11 (cont’d): Rational Expressions
Rational expressions are defined as the __________________of two polynomials.

Examples of Rational Expressions:

               4               x+5                  4x 2             3x − 2
                 ,                 ,                      ,
               x               x−5                 x2 − 9          x − 3x + 1
                                                                    2




Examples: Copy steps/answers from video.

      4                                                            x+5
1.                                                            2.
      x                                                            x−5



                       Fill in the blank(s):
      Definition

                       Principle of Zero Products
                          If a product is _____________, then one or more of the factors must be zero.



Copy steps/answers from video.

     4                                           2−x                               x+2
3.                                     4.                                  5.
     5x                                        3 ( x − 1)                        x − 5x + 6
                                                                                  2




     12                                                            168
6.                                                            7.
     40                                                            180




     3x − 15                                                       15x 2 − 20x
8.                                                            9.
       3x                                                           15x − 10




               Fill in the blank(s):

           To Simplify Algebraic Fractions:
                         1. Completely ___________ the numerator and denominator.
                         2. Cancel ________________factors.


                                                                                                         5
Copy steps/answers from video.

            2x − 8                                                                               2y 2 − 6y
10.                                                                                11.
         x + x − 20
          2
                                                                                            4y 3 + 16y 2 − 84y




                  STOP THE LESSON AND WORK THE PROBLEM SET

                                         Problem Set: Rational Expressions
Determine restricted values for the denominator.

          r 2 − r − 42                                         8                                                4m + 5
1.                                                  2.                                                  3.
         r 2 + 4r − 12                                       5x − 2                                             m2 − 5m




Simplify.

          r 2 − r − 42                                       6x 2 − 18x                                         2x 3 − 24x 2 + 54x
4.                                                  5.                                                  6.
         r 2 + 4r − 12                                        5x − 15                                            x 3 + 6x 2 − 27x




             RESUME THE LESSON FOR ANSWERS AND SOLUTIONS
     (Note: On the video you will first see the answers only. Following the answers, solution steps for all problems are also shown on the video.)




                  STOP THE LESSON AND WORK THE PROBLEM SET

                                                                  Self Quiz
Expand and multiply.
        3
    5
1a.                                                                              b.      (–3)4
    2




                                                                                                                                                     6
                               1
Evaluate for: a = −                  b = –4
                               3
2.         –a2 – b



Apply the correct Law of Exponents and simplify.

         −18xy                                                                                                      1 2 ( 3 )  1 4 
3.                                                  4.      (–3m2n)3                                    5.          − a b  8a b  a 
          9xy 2                                                                                                     4           2 




Find the products.
6.       4x3y(3x2y + 2xy – 7y)                              7.      (y – 10)(y + 9)                           8.      (r – 1)(r – 10)




Factor completely.
9.       20p2q – 10pq + 30p                         10.     x2 – 12x + 32                               11.     5p2 – 45p + 90




Identify restricted values for the denominator.
         13                                                                                  5x − 3
12.                                                                                13.
         5x                                                                                 x −x−2
                                                                                             2




Simplify.

           3p                                                                                   4y 2 + 4
14.                                                                                15.
         6p − 9                                                                             y 3 − 2y 2 − 3y




              RESUME THE LESSON FOR ANSWERS AND SOLUTIONS
     (Note: On the video you will first see the answers only. Following the answers, solution steps for all problems are also shown on the video.)


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