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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 x1.)
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:1 6
2 3
-1 -6
-2 -32. 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.