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Section 4-3
The Remainder and Factor Theorems
Remainder Theorem
Remainder Theorem – If a polynomial P(x) is divided by x-r, the remainder is a constant, P(r), and where Q(x) is a polynomial with degree one less than the degree of P(x).
Example 1
Synthetic Divisionby x+3
62 xxRemainder 13
534 23 xxx
Divide using synthetic division.
yyyy 234 246 By y+2
Divide using synthetic division.
yyyy 234 246 By y+2
- 2 -1 0 4 0 3 2 0 2 -4 0 0 -6 8 -1 2 0 0 3 -4 8
432 45 yyy Remainder of 8
Divide using synthetic division. By (x+2)8102 xx
Divide using synthetic division. By (x-1)143 23 xxx
Divide using synthetic division. By (x+1)533 xx
Divide using synthetic division. By (x-4)4372 234 xxxx
Example 3
Example 4
Use the remainder and factor theorem to find the remainder for each division. State whether
the binomial is a factor of the polynomialby (x+1)
822 xx1624 23 xxx
Use the remainder and factor theorem to find the remainder for each division. State whether
the binomial is a factor of the polynomialby (x+4)
822 xx8102 xx
Use the remainder and factor theorem to find the remainder for each division. State whether
the binomial is a factor of the polynomialby (x-1)
822 xx123 x
Use the remainder and factor theorem to find the remainder for each division. State whether
the binomial is a factor of the polynomialby (x-4)
822 xx164 24 xx
Example 5