**Part II:** Suppose you were given the function F(x) = x4 − 2×3 + 3×2 − 10x + 3 and the factor (x − 2). What is the value of a? **(1 point)**

**Part III:** Use the Remainder Theorem to explain whether or not (x − 2) is a factor of F(x) = x4 − 2×3 + 3×2 − 10x + 3. **(3 points)**

**3.** Factor the polynomial F(x) = x3 − x2 − 4x + 4 completely.

**Part I:** Find and list all the possible roots of F(x). Show your work. **(2 points)**

**Part II:** Use the Remainder Theorem to determine which of the roots from Part I are roots of F(x). Show your work. **(3 points)**

**Part III:** Factor the polynomial F(x) = x3 − x2 − 4x + 4 completely. Show your work. **(2 points)**

**Part IV:** Check your answer from Part III by multiplying the factors. Show your work. **(2 points)**

**4.** Find the roots of the polynomial equation H(x) = x3 − 7×2 − x + 7.

**Part I:** How many total roots must there be in this function? Explain how you know. **(1 point)**

**Part II:** Describe how many positive real roots are possible. **Hint:** Use Descartes’ rule of signs. **(1 point)**

**Part III:** Describe how many negative real roots are possible. **Hint:** Use Descartes’ rule of signs. **(2 points)**

**Part IV:** In light of the number of total roots for this function, how many possible complex roots are there? **(2 points)**

**Part V:** If −1 is one of the roots of H(x) = x3 − 7×2 − x + 7, use synthetic division to find the resulting quadratic expression. **Hint:** If −1 is a root, (x + 1) is the factor. **(2 points)**

**Part VI:** Factor the quadratic expression from Part V to find the final 2 factors and roots. **(2 points)**

**5.** Graph the polynomial equation H(x) = x3 − 7×2 − x + 7.

**Part I:** Describe the end behaviors of the graph of H(x). **(2 points)**

**Part II:** What can you say about the number of extreme values H(x) will have? **(2 points)**

**Part III:** Find the x-intercepts. **Hint:** Remember, one root was −1. You found the other two roots of H(x) in Question 4, Part VI. **(1 point)**

**Part IV:** Find the y-intercept and write it as a coordinate pair. **(1 point)**