Remainder and Factor Theorems

Remainder and Factor Theorems

Synthetic Substitution

Applying the Remainder Theorem using synthetic division to evaluate a function is called synthetic substitution.

Use synthetic substitution to find  $f\left(6\right)$  
  when  $f\left(x\right)=2x^4-5x^2+8x-7.$  

Factor Theorem
The binomial  $x-r$   is a factor of the polynomial  $P\left(x\right)$   iff  $P\left(r\right)=0.$  

• Can be used to determine whether a binomial is a factor of a polynomial.

• Can be used to determine all the factors of a polynomial.

Open Ended

Show that  $x+2$ is a factor of  $3x^3-4x^2-17x+6.$ Then find the remaining factors of the polynomial.

Open Ended

Show that  $x+1$ is a factor of  $x^5+x^4-5x^3-5x^2+4x+4.$ Then find the remaining factors of the polynomial.