Microsoft Word - PMA12_U3_LG.docx PC Math 12 XXXXXXXXXXPage 1 of 27 Name: UNIT 3 LEARNING GUIDE – POLYNOMIALS Instructions: Using a pencil, complete the following questions as you work through the...

Pre calculus 12, please show work (steps)


Microsoft Word - PMA12_U3_LG.docx PC Math 12 2021-11-03 Page 1 of 27 Name: UNIT 3 LEARNING GUIDE – POLYNOMIALS Instructions: Using a pencil, complete the following questions as you work through the related lessons. Show ALL work as it is explained in the lessons. Do your best and ask your instructor if you do not understand any questions! 3.1 FACTORING REVIEW 1. Factor each polynomial. a) 4?!? + 20?? − 16?! b) 5?!? − 10?? − 20?! c) 6?ℎ! + 15?!ℎ" − 9?ℎ# d) −40?!?# − 8?!?! − 16?"?# e) ?(3? + 1) − 2(3? + 1) f) ?(? − 4) + 8(? − 4) 2. . Factor each trinomial a) ?! + 10? + 16 b) ?! + 12? + 35 c) ?! + ? − 2 d) ?! − 6? + 8 e) ?! + 4? − 12 PC Math 12 2021-11-03 Page 2 of 27 3. Factor the following polynomials a) 4?! − 1 b) 36?! − ?! c) 25?! − 144ℎ! d) 625 − ?! e) 2500 − 81?! f) ?# −?! 4. Utilize all of the strategies for factoring in order to factor the following polynomials a) ?! − 4? − 2? + 8 b) −10? + ?! − 25 + 10? c) (2?! + 23?) − (3? − 18) d) 5?? + 3??! − 2?!? − 7??! + 3?? e) 4?! + 10 + ? − 50 − ?! + 4 + 11? f) −100 + 9?! + 19 − 5?! g) −7? + 5?! + 6 − 3?! − 11? + 30 h) 2?? + 30?! − ?? − 16?# − ?? − 5?! 5. Utilize all of the strategies for factoring in order to factor the following polynomials. a) ?! + 2? + 1 − ?! b) 4?! − ?! − 10? − 25 c) ?" + 2?! + 6? + 12 d) ?! − 5? + 3?? − 15? PC Math 12 2021-11-03 Page 3 of 27 3.2 REVIEW MESSY TRINOMIALS 1. 1. Factor each of the following completely. Show your work using the procedural method. Verify the factoring is correct by mentally multiplying the factors a) 2x2 + x – 1 b) 2x2 + 11x + 12 c) 4x2 + 9x + 2 d) 6x2 – 13x + 6 e) 6x2 – 17x + 10 f) 12x2 + x – 6 g) 4x2 – 16x + 15 h) 12x2 – 5x – 25 PC Math 12 2021-11-03 Page 4 of 27 2. Factor each of the following completely. Show your work using the procedural method if the trinomial is messy. a) 4x2 + 14x + 6 b) 8x2 + 28x – 16 c) 12x2 + 24xy + 9y2 d) $ ! x2 + % ! x + 6 e) 3x2 – 3x + ! " f) 0.1x2 + 0.7x + 1 3. Factor each of the following completely. It may be necessary to rearrange the terms first. a) –14 – 15x – 4x2 b) 34x + 24x2 + 7 PC Math 12 2021-11-03 Page 5 of 27 c) 2x4 + 7x2 + 6 d) 8x4 – 2x2 – 3 e) 3x4y2 + 10x2y + 8 f) 10x4 + 37x2y + 7y2 PC Math 12 2021-11-03 Page 6 of 27 3.3 SOLVING BY FACTORING 1. Solve each of the following by factoring. a) m2 + 2m = 0 b) m2 + 9m + 18 = 0 c) b2 + 12b + 32 = 0 d) 7p2 – 19p – 6 = 0 e) 5n2 – 6n + 1 = 0 f) 7p2 + 39p – 18 = 0 PC Math 12 2021-11-03 Page 7 of 27 2. Solve each of the following by factoring. You will need to rearrange the terms first to create a trinomial. a) 7b2 + 70b + 176 = 8 b) k2 – 10k = –16 3.4 THE REMAINDER THEOREM 1. In your own words explain the Remainder Theorem. 2. Using either long or synthetic division, determine the division statement for the following questions. a) (x3+4x2−6x−9)÷(x−2) b) (7x3−9x2+3x)÷(x−1) PC Math 12 2021-11-03 Page 8 of 27 3. Use the remainder theorem to determine the remainder when each of the following polynomials is divided by the binomial (x+4). a) P(x)=x2−3x+5 b) P(x)=3x2+10x−8 c) P(x)=x3−7x+6 d) P(x)=−x3+5x2−x+12 e) Which of the above polynomials has a root of –4? Explain 4. Use the remainder theorem to determine the remainder for each of the following divisions. a) (x2−7x)÷(x+6) b) (x2+4x+9)÷(x−m) c) (x3−x+4)÷(x+2) d) (2m3+3m2−5m−10)÷(2m−1) 5. a) The volume of an open top box is given by V(x)=x3+2x2−5x−6and the length can be expressed as (x+1). Determine the factors that describe the width and height. Assume the width is less than the height. PC Math 12 2021-11-03 Page 9 of 27 b) Whatarethedimensionsoftheboxifxequals7ft? 6. Given the remainder, determine the value of m for each of the following operations. a)P(x)=(x2+mx−7)÷(x−1); remainder = 5 b) P(x)=(x3+mx2+6x–3)÷(x+3); remainder = –21 c) P(x)=(x4−3x3+x2+mx+5)÷(2x−1); remainder = 47 16 PC Math 12 2021-11-03 Page 10 of 27 7. When the polynomial P(x)=2x3+mx2+nx−5is divided by (x−1)the remainder is 1. When the polynomial is divided by (x+2)the remainder is –5. Determine the values of both m and n. 8. P(x)=7x3+mx2+nx+2. When P(x) is divided by (x + 1), the remainder is 6. When P(x) is divided by (x – 2), the remainder is 42. Determine the values of both m and n. PC Math 12 2021-11-03 Page 11 of 27 3.5 THE FACTOR THEOREM 1. Explain in your own words how to determine all the potential rational roots for a polynomial and then do so for the polynomials below. a) P(x)=x3+2x2−7x−4 b) P(x)=2x3−5x2+3 c) P(x)=−3x3+2x2−7x+6 2. Could (x−3)be a factor of the polynomial P(x)=2x3+6x+5? Why or why not? Show your work. PC Math 12 2021-11-03 Page 12 of 27 3. Use the factor theorem to determine whether each of the following polynomials has