Section 3.1 Remainder Theorem and Factor Theorem
Division Algorithm:
Polynomial Division (Long and Synthetic):
• Arrange polynomials in order of descending powers
• put in "0s" for any "missing" terms to hold the places
• divide
Long Division:
• Divide 1st term into the 1st term
• Multiply divisor by quotient
• Subtract (distribute the ­ sign first)
• Bring down next term
• Repeat as needed
• Write the remainder/divisor
Example:
Synthetic Division ­ works when the divisor is linear in the form (x­c) or (x +c)
We temporarily remove the variable and work with the coefficients. Replace the variables reduced by one power at the end of the problem.
• Use opposite sign of c in the "corner"
• Write coefficients from terms in order of descending powers across top level
• Be sure to use a "0" for any "missing" terms
• Bring down the first coefficient under the "shelf"
• multiply by corner
• write on the shelf under next coefficient
• ADD and write under the shelf
• Repeat
• The last number is the remainder
• Replace the variable reducing the power by 1.
Examples:
Remainder Theorem: If a polynomial P(x) is divided by x ­ c, then the remainder equals P(c).
Proof:
EXAMPLES: Given P(x) = 3x3 + 4x2 - 5x + 3, use the remainder theorem
to find P(c) for
a) c = -4
b) c = 3
Factor Theorem: • A polynomial P(x) has a factor (x ­ c) if and only if P(c) = 0. That is, • (x ­ c) is a factor of P(x) if and only if c is a zero of P.
EXAMPLE: Use synthetic division and the Factor Theorem to
determine whether (x + 5) or (x - 2) is a factor of
P(x) = x4 + x3 -21x2 - x + 20
EXAMPLE: Use the information from the above example to write P(x)
as a product of a linear factor and a reduced polynomial
SUMMARY: In the division of the polynomial P(x) by (x ­ c), the remainder is
•
•
•
•
equal to P(c).
0 if and only if (x ­ c) is a factor of P(x).
0 if and only if c is a zero of P.
0 if and only if (c,0) is an x­intercept of the graph of P.