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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.) 7