Example 1 divisor dividend quotient remainder Remainder Theorem: The remainder is the value of the function evaluated for a given value.
Example 2 – Synthetic Division (All exponents must be accounted for; zero is inserted in for x 1.) 1 1. Drop the first coefficient down. 2.Multiply by the divisor & place under next coefficient, then add. - 1 - 2 3. Repeat with all coefficients. 2 2 - 2 0 4. Place a vertical bar before the last value, separating the quotient from the remainder. 5. The values to the left of the bar are the coefficients of a polynomial of a degree one less than the original polynomial. Factor Theorem: If the remainder of this process is zero, the quotient is a factor of the original polynomial. Meaning it is one polynomial that was multiplied with others resulting in the original polynomial. (Another name for the quotient is a “depressed polynomial”)
Example 3 Long division or synthetic division could also be used. If the remainder is zero, then x – 1 is a factor. Since f(1) = 0, x – 1 is a factor of the polynomial.
Example 4 (without graphing!) 1. List all the factors of the constant: 16231623 -1 -6 -2 -3 2. Evaluate each factor in the polynomial: Because the polynomial is a degree of three, there should be three binomial factors. These binomial factors consist of the constant factors that resulted in a value of zero: 1, 2, -3, so the binomial factors are: (x – 1), (x – 2), & (x – 3) You can verify the results by multiplying it out.