2.2 DIVIDING POLYNOMIALS; REMAINDER AND FACTOR THEOREMS

REVIEW OF FACTORS

What is a factor?

How do you know if 4 is a factor of 50?

Is 4 a factor of 124?

LONG DIVISION REVIEW

1

88 97

116

1

2

7

r1

LONG DIVISION OF POLYNOMIALS

EXAMPLE

Divide using long division. State the quotient, q(x), and the remainder, r(x).

(6x³ +17x²+ 27x + 20) (3x + 4)

EXAMPLE

Divide using long division. State the quotient, q(x), and the remainder, r(x).

24 8 6 2 1x x x

REMAINDERS CAN BE USEFUL!

THE REMAINDER THEOREM: If the polynomial P(x) is divided by (x – a), then the remainder is P(a).

SYNTHETIC DIVISION Quick method of dividing polynomials

Used when the divisor is of the form x – a

Last column is always the remainder

EXAMPLE

Divide using synthetic division.

3 2 1x x x

EXAMPLE

Divide using synthetic division.5 4 3 22 3 1

2

x x x x x

x

FACTOR THEOREM For a polynomial P(x), x-a is a factor if an only if

P(a)=0

Or in other words, If f(c) = 0, then x – c is a factor of f(x). If x – c is a factor of f(x), then f(c) = 0.

If we know a factor, we know a zero! If we know a zero, we know a factor!

PG. 61 # 20

You are given the polynomialand one of the roots are x= -2. Find the other roots.

PG. 61 # 22

You are given the polynomial

two of the roots are x= -2 and x=3 . Find the other roots.