# Part II: Suppose you were given the function F ( x ) = x 4 2 x 3 + 3 x 2 10 x + 3 and the factor ( x 2). What is the value of a ?

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)