Roots of Polynomials Test Review

Factor the polynomial and state the roots.  $x^3+2x^2+4x+8=0$  

•  $\left\{2,\ 2i,\ -2i\right\}$  

•  $\left\{-2,\ 2i,\ -2i\right\}$  

•  $\left\{-2,\ 2,\ -2i,\ 2i\right\}$  

•  $\left\{2,\ 2i\right\}$  

Factor each polynomial and state the roots.  $x^4-12x^2+32=0$  

•  $\left\{2,\ \ 2\sqrt{2}\right\}$  

•  $\left\{2,\ -2,\ 2\sqrt{2}\right\}$  

•  $\left\{2,\ -2,\ \sqrt{8}\right\}$  

•  $\left\{2,-2,\ 2\sqrt{2},-2\sqrt{2}\right\}$  

Write a polynomial function of least degree in standard form that has the follow roots.
 $-3,\ \sqrt{5}$  

•  $f\left(x\right)=\left(x+3\right)\left(x^2-5\right)$  

•  $f\left(x\right)=\left(x-3\right)\left(x-\sqrt{5}\right)$  

•  $f\left(x\right)=x^3+3x^2-5x-15$  

•  $f\left(x\right)=x^3-3x^2+5x+15$  

Write a polynomial function of least degree in standard form that has the follow roots.
 $3,\ -3i$  

•  $f\left(x\right)=\left(x-3\right)\left(x+3i\right)$  

•  $f\left(x\right)=x^3-3x^2+9x-27$  

•  $f\left(x\right)=x^3+3x^2+9x-27$  

•  $f\left(x\right)=\left(x-3\right)\left(x-3i\right)$  

Use the graphing calculator to find the number and type of roots of the polynomial function.  $f\left(x\right)=x^5+5x^4-x^3-5x^2-20x-100$  

• 5 total roots
  1 real rational root at -5
  2 real irrational roots at  $\pm2.236$  
  2 non-real complex roots 

• 5 total roots
  1 real rational root at -5
  1 real irrational root at  $2.236$  
  2 non-real complex roots

• 5 total roots
  3 real rational roots at -5, -2.236, 2.236
  
  0 real irrational roots
  
  2 non-real complex roots

• 4 total roots
  1 real rational root at -5
  2 real irrational roots at  $\pm2.236$  
  2 non-real complex roots

Use the graphing calculator to find the number and type of roots of the polynomial function. 
 $f\left(x\right)=x^4-5x^2-14$  

• 4 total roots
  2 real rational roots at  $\left\{-7,\ 7\right\}$  
  2 non-real complex roots

• 4 total roots
  
  1 real rational root at  $\left\{-7\right\}$  
  1 real irrational root at  $\left\{\sqrt{7}\right\}$  
  2 non-real complex roots

• 4 total roots
  2 real irrational roots at  $\pm\sqrt{7}$  
  2 non-real complex roots 

• 4 total roots
  
  2 real rational roots at  $\left\{\pm17\right\}$  
  2 real irrational roots at  $\left\{\pm\sqrt{7}\right\}$  

Find all exact roots.
 $f\left(x\right)=x^3-4x^2+5x-2$  

•  $\left\{2,\ 1\right\}$  

•  $\left\{-2,\ 2,\ -1,\ 1\right\}$  

•  $\left\{2,\ 1\ mult.\ 2\right\}$  

•  $\left\{2,\ 1,\ 1\right\}$  

Find all exact roots.  $f\left(x\right)=x^3+7x^2-7x-57$  

•  $\left\{-3,\ -2\ \pm\ \sqrt{23}\right\}$  

•  $\left\{-3,\ -2,\ \pm\sqrt{23}\right\}$  

•  $\left\{3,\ -3,\ 2,\ -2,\ \pm\sqrt{23}\right\}$  

•  $\left\{-3,\ \pm\sqrt{23}\right\}$  

Find all exact roots.  $x^5-81x=0$   

•  $\left\{0,\ 3,\ 3i\right\}$  

•  $\left\{0,\ 3,\ \pm3i\right\}$  

•  $\left\{\pm3,\ \pm3i\right\}$  

•  $\left\{0,\ \pm3,\ \pm3i\right\}$  

Find all exact roots.  $3x^3+18x^2+22x+35=0$   

•  $\left\{5,\ -3,\ \pm\frac{5i\sqrt{3}}{6}\right\}$  

•  $\left\{-5,\ \frac{-3\pm5i\sqrt{3}}{6}\right\}$  

•  $\left\{-5,\ \frac{-3+5i\sqrt{3}}{6}\right\}$  

•  $\left\{5,\ \frac{-3\pm5i\sqrt{3}}{6}\right\}$