Practice 2 Solutions

1. Explain if it is possible for any polynomial to have an odd number of non-real, complex roots.

Because non-real, complex roots come in conjugate pairs it is not possible to have an odd-number.

2. Describe what is happening to the x- and y-values when the function decreases across an interval.


As the x-values increase, the y-values decrease.

3. Explain why it is important to identify the turning points on the graph of a polynomial function.


Turning points are used to find the relative minimum and maximum across an interval. Turning points are also used to identify increasing and decreasing intervals.

4. Describe the Fundamental Theorem of Algebra.

The Fundamental Theorem of Algebra says that an $n^{th}$ degree polynomial will have n complex roots.

5. A seventh degree polynomial function where $a<0$, with five real roots

Sample sketches:

6. A fourth degree polynomial function where $a>0$ and a real root with a multiplicity of two

Sample sketches:

7. A third degree polynomial with a positive leading coefficient, and a non-real, complex conjugate pair

Sample sketches:

8. A sixth degree polynomial with a negative leading coefficient, and four real roots

Sample sketches:

9. $h\left(x\right)=–\left(x^2+4\right)\left(x^2–x–2\right)$

$\begin{array}{ccc}{x}^2+4=0& & \left(x+1\right)\left(x–2\right)=0\\ x^2=–4& & x=–1, 2\\ x=\pm 2i& & \end{array}$

1. 
2. $x=\pm 2i, –1, 2$

10. $m\left(x\right)=\left(3x^3–x^2\right)\left(x^2+6\right)$

 $$\begin{gathered} \left(3x^3–x^2\right)\left(x^2+6\right) \\ \begin{array}{ccc}{x}^2\left(3x–1\right)& & x^2=–6\\ x=0, \frac{1}{3}& & x=\pm i\sqrt{6}\end{array} \end{gathered}$$

1. 
2. $x=\pm i\sqrt{6}, 0, \frac{1}{3}$

11. $f\left(x\right)=3x^4–10x^3–24x^2–6x+5 \mathrm{across} \left[–2, 0\right]$

1. 
2. Increasing intervals: (–1, –0.137), (3.637, +∞)

   Decreasing intervals: (–∞, –1), (–0.137, 3.637)

3. Relative minimum: (–1, 0)

   Relative maximum: (–0.137, 5.398)

12. $g\left(x\right)=\left(x+2\right)\left(x+3\right)\left(x–1\right) \mathrm{across} \left[–0.5, 2\right]$

1. 
2. Increasing intervals: (–∞, –2.535), (–0.131, +∞)

   Decreasing intervals: (–2.535, –0.131)

3. Relative minimum: (–0.131, –6.065)

   Relative maximum: none

13. $h\left(x\right)=x^4+2x^3–8x^2 \mathrm{across} \left[–6, 1\right]$

1. 
2. Increasing intervals: (–2.886, 0), (1.386, +∞)

   Decreasing intervals: (–∞, –2.886), (0, 1.386)

3. Relative minimum: (–2.886, –45.335) 
   Relative maximum: (0, 0)

14. $q\left(x\right)=x^3+4x^2–5 \mathrm{across} \left[–4, 3\right]$

1. 
2. Increasing intervals: (–∞, –2.667), (0, +∞)

   Decreasing intervals: (–2.667, 0)

3. Relative minimum: (0, –5)

   Relative maximum: (–2.667, 4.481)