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.

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Applying the Remainder Theorem using synthetic division to evaluate a function is called synthetic what?

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

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot A.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot B.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot C.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot D.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot E.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot F.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot G.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot H.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot I.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot J.

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When finding  $f\left(6\right)$ when  $f\left(x\right)=2x^4-5x^2+8x-7$ using synthetic division, what number goes in spot K.

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K is the

,

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 $f\left(6\right)=$  

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If  $f\left(x\right)=2x^3-3x^2+7,$ find  $f\left(3\right).$     

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If  $f\left(x\right)=3x^3-6x^2+x-11,$  then  $f\left(3\right)=$  

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If  $g\left(x\right)=4x^5+2x^3+x^2-1,$   find  $f\left(-1\right).$  

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.