Polynomial Theorems Check-In

Use the Remainder Theorem to evaluate the following polynomial:

 $-x^3+6x-7$  at  $x=2$  

•  $f\left(2\right)=1$  

•  $f\left(-3\right)=2$  

•  $f\left(2\right)=-3$  

•  $f\left(2\right)=13$  

Find all the possible rational roots for the following polynomial:

 $f\left(x\right)=4x^3-3x^2-3x+2$  

•  $\frac{1}{4},\frac{1}{2},1,2$  

•  $\pm\frac{1}{2},\pm1,\pm2,\pm4$  

•  $\pm1,\pm2$  

•  $\pm\frac{1}{4},\pm\frac{1}{2},\pm1,\pm2$  

Find all the rational zeros for the following polynomial: Mark all that apply.


 $f\left(x\right)=2x^3-5x^2+4x-1$  

•  $-1$  

•  $1$  

•  $2$  

•  $\frac{1}{2}$  

•  $-\frac{1}{2}$  

Is $\left(x-4\right)$  a factor of $x^3+x^2-16x-16$ ?

• YES

• NO

Which of the following is a COMPLETE list of all possible Rational Zeros?   $f\left(x\right)=3x^3+2x^2-6x+7$  

•  $\pm1,\ \pm7,\ \pm\frac{1}{3},\ \pm\frac{7}{3}$  

•  $\pm1,\ \pm7$  

•  $\pm1,\ \pm3,\ \pm7,\ \pm\frac{7}{3}$  

•  $\pm1,\ \pm3,\ \pm7,\ \pm\frac{1}{3},\ \pm\frac{7}{3}$  

According to the Rational Root Theorem, what are the all possible rational roots?    $2x^3-11x^2+12x+9=0$  

•  $\pm1,\ \pm2$  

•  $\pm1,\ \pm3,\ \pm9$  

•  $\pm1,\ \pm3,\ \pm9,\ \pm\frac{1}{2},\ \pm\frac{3}{2},\ \pm\frac{9}{2}$  

•  $\pm1,\ \pm2,\ \pm\frac{1}{3},\ \pm\frac{2}{3},\ \pm\frac{3}{2},\ \pm\frac{2}{9}$  

The degree of the polynomial determines the number of roots.

• True
• False

How many possible negatives solutions does the following polynomial have?   $f\left(x\right)=x^4-3x^3-17x^2+39x-20$  

• 3 or 1

• 3

• 1

• 2 or 0

 How many positive solutions does the following polynomial have?  $f\left(x\right)=3x^5-x^4+3x^2-6x+11$  

• 4, 2, or 0

• 4

• 3 or 1

• 3

Using Descartes' Rule of Signs, how many possible NEGATIVE roots are there for the polynomial   $f\left(x\right)=x^4+x^3+x^2+x+5$ .  Select all that apply.

• 0

• 1

• 2

• 3

• 4