Remainder and Factor Theorems

Use synthetic division or long division.

 $\frac{x^3-x^2+4}{x+2}$  

•  $x^2+3x+6+\frac{-8}{x+2}$  

•  $x^2-3x+6+\frac{-8}{x+2}$  

•  $x^2-2x+6+\frac{6}{x+2}$  

•  $x^2-3x+6+\frac{6}{x+2}$  

Use synthetic division or long division.

 $\frac{-5x^3-20x^2-19x-12}{x+3}$  

•  $5x^2-3x-4$  

•  $-5x-5$  

•  $-\frac{5}{3}x^2-\frac{20}{3}x-\frac{19}{3}$  

•  $-5x^2-5x-4$  

One of the zeros of

 $f\left(x\right)=x^3-4x^2-25x+100$  is 5.  Factor  $f\left(x\right).$  

•  $\left(x+4\right)\left(x-5\right)\left(x+5\right)$  

•  $\left(x-5\right)\left(x-4\right)\left(x+4\right)$  

•  $x\left(x-4\right)\left(x-5\right)$  

•  $\left(x-4\right)\left(x-5\right)\left(x+5\right)$  

One of the zeros of

 $f\left(x\right)=x^3-48x-128$  is  $-4$ .  Factor  $f\left(x\right).$  

•  $\left(x+4\right)^2\left(x-8\right)$  

•  $\left(x+4\right)^2\left(x+8\right)$  

•  $\left(x+4\right)\left(x-4\right)\left(x-8\right)$  

•  $\left(x-4\right)^2\left(x-8\right)$  

Use the Factor Theorem to decide if  $\left(x-3\right)$  is a factor of  $g\left(x\right)=x^3-5x^2+11x-15.$  
 

• Yes

• No

Use the Factor Theorem to decide if  $\left(x-2\right)$  is a factor of  $g\left(x\right)=4x^2-2x+20.$  
 

• Yes

• No

What is the remainder of $\left(3x^3-6x^2-3x+10\right)\div\left(x+2\right)$ ?

What is the remainder of $\left(4x^5+2x^3-4x^2+4\right)\div\left(x+3\right)$ ?

Use long division to divide.

 $\frac{x^4+7x^3+17x^2+20x}{x^2+4}$  

•  $x^2+13x+7+\frac{-52}{x^2+4}$  

•  $x^2+7x+\frac{13x-8}{x^2+4}$  

•  $x^2+7x+13+\frac{-8x-52}{x^2+4}$  

•  $x^2+1+\frac{1}{x^2+4}$  

If $\left(x-2\right)$  is a factor of  $4x^3-23x^2+26x+8$  , what are the zeros?

•  $\left\{\frac{1}{4},2,-4\right\}$  

•  $\left\{1,-1,2\right\}$  

•  $\left\{-1,-1,-2\right\}$  

•  $\left\{-\frac{1}{4},2,4\right\}$