Polynomial Theorems

Use the Remainder Theorem to evaluate the following polynomial:

 $-x^3+6x-7$  at  $x=2$  

•  $f\left(2\right)=1$  

•  $f\left(-3\right)=2$  

•  $f\left(2\right)=-3$  

•  $f\left(2\right)=13$  

A polynomial function with rational coefficients has the following zeros. 

 $-1,\ 1+3i$  

Determine all the additional zeros.

•  $1-3i$  

•  $-3i$  

•  $1$  

• No additional zeros

Which is a possible polynomial function the is represented by the following zeros:

 $\frac{4}{5},\ 2i$  

•  $\left(5x+4\right)\left(x+2i\right)$  

•  $\left(5x+\text{4}\right)\left(x^2-4\right)$  

•  $\left(5x+4\right)\left(x^2+4\right)$  

•  $\left(4x+5\right)\left(x^2+4\right)$  

Explain why it makes sense that a third degree polynomial must have at least one rational zero.

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.


 $f\left(x\right)=2x^3-5x^2+4x-1$  

•  $-1$  

•  $1$  

•  $2$  

•  $\frac{1}{2}$  

•  $-\frac{1}{2}$