Long Division can be used to divide a polynomial by a binomial.
The result of the division of a polynomial function P(x) by a binomial of the form x – b can be
written as P(x) = (x-b)Q(x) + R where Q(x) is the quotient and R is the remainder.
Division Statement: divisor x quotient + remainder = dividend
can be used to check the result of a division
Remainder theorem states that when a polynomial function P(x) is divided by x – b,
the remainder is P(b), and when it is divided by ax-b, the remainder is P(b/a), where a
and b are integers and a not = 0.
For integer values of a and b, with a not equal 0,
Factor Theorem states that x – b is a factor of a polynomial P(x) if and only if P(b) = 0.
Similarly, if ax – b is a factor of P(x) if and only if P(b/a) = 0
Integral Zero Theorem states that if x – b is a factor of a polynomial function P(x) with
leading coefficient 1 and remaining coefficients that are integers, then b is a factor of the
constant term P(x).
Rational Zero Theorem states that if P(x) is a polynomial function with integer coefficients
and x = b/a is a rational zero of P(x), then
b is a factor of the constant term of P(x)
a is a factor of the leading coefficient of P(x)
ax – b is a factor of P(x)
Real roots of a polynomial equation P(x) = 0 correspond to the x-intercepts of the graph of the
polynomial function P(x).
X-intercepts of the graph