Pre-Calculus 12 Notes

By: Gen L.

Lesson 2: Substitution and Remainder Theorem

Substitution

If , what is the value of ?

Division

Determine the remainder when is divided by

• Wait, it's the value of f(2)
• Yep. This means that

Comparison

They look the exact same!

Remainder / Modulus Theorem

let be a polynomial. For

• In plain terms, The remainder when dividing a polynomial by is equal to the value of the polynomial evaluated at .
• This means Synthetic Substitution is equivalent to Synthetic Division.

Example

Use the remainder theorem to determine the remainder of divided by .

Q and R

Find the quotient and remander when divided by

• If the remainder is zero: The divisor divides evenly, which means it's a factor. (you'll see more later.)

Finding Coefficients

Find and if

Here's the process:

Finding Coefficients: Step 1

1. Evaluate for each.

Finding Coefficients: Step 2

2. Solve this system.
   
   

• We'll solve for first.
  
  
• Now, solve for .
  
  

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