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All of our algebra lives we have learned to factor over the integers. This gave us techniques like: Greatest Common Factors, X Marks the Spot, Difference of Squares and Polynomials of Quadratic Type. Once we acknowledged the relationship between the zeros of a polynomial and the factors of a polynomial, this opened the door to factoring over the reals. The Fundamental Theorem of Algebra tells us that every polynomial factors into linear factors, irreducible quadratics and leading coefficients. The Factor Theorem allows us to use equation techniques like Square Roots and The Quadratic Formula to find linear factors of polynomials. Furthermore we have added The Rational Roots Theorem to our toolbox. With all of these tools at our disposal we encounter a new type of problem: "Which technique should I use?" The answer is practice and intuition. Perhaps most importantly, we must remember that when we try to hammer in a screw, once we see it isn't going to work, we can go look for that screwdriver. I. Factoring Over the Reals Factor each of the polynomials below over the Reals. (60 points) 1) x2 + 4x - 32 2) x2 - 8 3) x2 - 6x + 7 4) x2 - 23x + 120 5) 3x2 + 6x - 18 6) 2x2 - 6x + 1 7) x3 - 3x2 + 3x - 1 8) x3+ 70x2 + 1200x 9) x3 + 2x2 - 1 10) x3 - x2 - 10x - 8 11) 8x3 + 36x2 + 46x + 15 12) x4 - 16x2 + 64 II. Self-Evaluation For each polynomial state which technique(s) you used. Explain why you chose those techniques. (24 points) III. Self-Reflection Please answer the following questions with a short paragraph. Answer these questions individually. (16 points) a) What is a skill you have that really benefited your lab group? b) What did you learn from this lab? c) Describe a mistake you made while working on this lab and how your group helped you. d) What about your thinking, learning or work on this lab brought you the most satisfaction? Why