Algebra 2 Honors
Section 5.6 Notes: The Remainder and Factor Theorems
Synthetic division can be used to find the value of a function. Consider the polynomial function f(x) = –3x2 + 5x + 4. Divide the
polynomial by x – 3.
Now find f(3) for f(x) = –3x2 + 5x + 4
How are the two related?
Divide f(x) = x3 – 3x2 – 7x + 6 by x + 2
Find f(-2) 
What do you notice?
Notice that the value of f(a) is the same as the remainder when the polynomial is dived by x – a. This illustrates the Remainder
Theorem.
Applying the Remainder Theorem using synthetic division to evaluate a function is called synthetic substitution.
Example 1: If f(x) = 2x4 – 5x2 + 8x – 7, find f(6).
Example 2: If f(x) = 2x3 – 3x2 + 7, find f(3).
Example 3: The number of college students from the United States who study abroad can be modeled by the function
S(x) = 0.02x 4 – 0.52x 3 + 4.03x 2 + 0.09x + 77.54, where x is the number of years since 1993 and S(x) is the number of students in
thousands. How many U.S. college students will study abroad in 2011?
The synthetic division below shows the quotient of 2x3 – 3x2 – 17x + 30 and x + 3 is 2x2 – 9x + 10.
When you divide a polynomial by one of its binomial factors, the quotient is called a depressed polynomial. A depressed polynomial
has a degree that is one less than the original polynomial.
In the example above, the remainder is 0, this means that x + 3 is a factor of 2x3 – 3x2 – 17x + 30. This illustrates the Factor
Theorem, which is a special case of the Remainder Theorem.
*If the remainder is 0, then (x – k) is a factor of the polynomial.
*k is called a zero because f(k) = 0.
*Same format as synthetic substitution. Use FILLERS!
*Used when divisor is in the form x – k.
*Create a polynomial out of coefficients that are left under the line. Factor further, if possible.
Example 4: Determine whether x – 3 is a factor of x3 + 4x2 – 15x – 18. Then find the remaining factors of the polynomial.
Example 5: Determine whether x + 2 is a factor of x3 + 8x2 + 17x + 10. If so, find the remaining factors of the polynomial.
Example 6: Factor f(x) = 3x3 + 13x2 + 2x – 8 given that f(–4) = 0.
Example 7: Given x3 – 7x2 + 4x + 12, if one of its factors is x – 2, find the remaining factors of the polynomial.
Example 8: Factor f(x) = 18𝑥 3 + 9𝑥 2 – 2x – 1 given that (2x + 1) is a factor.
Example 9: Use the graph of the given function to factor completely.
f ( x)  x3  8x 2  9 x  18