DIVIDING POLYNOMIALS
(Long & Synthetic Division)
LONG DIVISION RECALL
3 137 5
Division Statement: dividend = (divisor)(quotient) + remainder
POLYNOMIAL DIVISION – Polynomial by a Binomial
Ex.
(–5x2 + 3x3 – 1 – 7x) ÷ (x – 3)
Place term in quotient above the term
of the dividend with the same degree!
x 3
Terms of the divisor &
dividend must be in
descending order
of degree.
(Check by re-expanding the division statement!!)
POLYNOMIAL DIVISION – Synthetic Division
Synthetic division is an efficient way to divide a polynomial by a binomial
of the form (x – k). (NOTE: the divisor must be a binomial!!)
Ex.
(–5x2 + 3x3 – 1 – 7x) ÷ (x – 3)
Ex.
(13x – 2x3 + x4 – 6) ÷ (x + 2)
Zero must be used as
the coefficient of any
missing powers of
the variable!!
POLYNOMIAL DIVISION – Polynomial by a Trinomial
Ex.
(5x – 2x3 + 3 + x4) ÷ (1 + 2x + x2)
Must use long division!!
USING DIVISION to FACTOR POLYNOMIALS
How do you know whether a divisor is a factor of the dividend??
Ex.
(2x + 3) is a factor of f(x) = 6x3 + 5x2 – 16x – 15.
Determine the other factors.
The quotient must be
divided by the common
factor removed
from the divisor!!
SUMMARY
 A polynomial can be divided by a polynomial of the same degree or less.
 Synthetic division is a more efficient version of polynomial division.
It can only be used to divide a polynomial by a binomial.
 When dividing polynomials:
 terms should be arranged in descending order of degree (divisor & dividend)
 zero must be used as the coefficient of any missing powers of the variable
 If the remainder of the division is zero, both the divisor and the quotient are
factors of the dividend.
Homework: p.169–170 #5ace, 6ace, 7ac, 8ac, 9a, 10ac, 11, 12a, 17
HINT: set up division statement!!