2.3
Polynomial Division
and
Synthetic Division
Ex. Long Division
6x2 - 7x
+ 2
3
2
x ! 2 6 x ! 19 x + 16 x ! 4
- 6x3 +- 12x2
- 7x2 + 16x - 4
2
+- 7x + 14x
2x - 4
2x - 4
What times x
equals 6x3?
Change the signs
and add.
6x3 – 19x2 + 16x – 4 = (x – 2) (6x2 – 7x + 2) + 0
f(x) = d(x)q(x) + r(x)
Dividend Divisor Quotient Remainder
Divide x3 – 1 by x - 1
x2 +
x +
3
2
1
x !1 x + 0x + 0x !1
- x3 +
-
x2
x2 + 0x - 1
x2 -+ x
x- 1
x- 1
Synthetic Division
Use synthetic division to divide x4 – 10x2 – 2x + 4
by x + 3. First, write the coef’s.of the dividend.
Put zeros in for missing terms.
-3
1
0
-10
-2
4
1
-3
-1
1
1
quotient
x3
-
3x2
- x + 1
1
+
x+3
Bring down the
1, mult. then
add diagonally.
remainder
Remainder Theorem: If a polynomial f(x) is
divided by x – k, then the remainder is
r = f(k)
Use the remainder theorem to find f(-2) if
f(x) = 3x3 + 8x2 + 5x - 7.
-2
3
8
5
-7
3
2
1
-9
f(-2) = -9
This means that (-2, -9) is a point on the graph of f.
Factor Theorem: A polynomial f(x) has a factor
(x – k) if and only if f(k) = 0.
Show that (x – 2) and (x + 3) are factors of
f(x) = 2x4 + 7x3 – 4x2 – 27x - 18
2
-3
2
2
2
7
11
5
-4
18
3
-27
9
0
(x – 2)(x + 3)(2x2 + 5x + 3)
-18
0
f(2) = 0
f(-3) = 0