Download Polynomial and Synthetic Division 2.3

Warm-Up
Use long division to divide 5 into 3462.
692
5 3462
30
46
45
12
10
2
Warm-Up
Use long division to divide 5 into 3462.
Divisor
692
5 3462
30
46
45
12
10
2
Quotient
Dividend
Remainder
Warm-Up
Use long division to divide 5 into 3462.
Dividend
Divisor
3462
2
 692 
5
5
Quotient
Remainder
Divisor
Remainders
If you are lucky enough to get a remainder of
zero when dividing, then the divisor divides
evenly into the dividend
This means that the divisor is a factor of the
dividend.
For example, when dividing 3 into 192, the
remainder is 0. Therefore, 3 is a factor of 192.
5-3 Dividing Polynomials
Skills:
•
•
Divide polynomials using long division.
Divide polynomials using synthetic division.
Glencoe – Algebra 2
Chapter 5: Polynomials
5
Vocabulary
As a group, define each
of these without
your book. Give an
example of each
word and leave a bit
of space for
additions and
revisions.
Quotient
Remainder
Dividend
Divisor
Divides Evenly Factor
Two Types of Polynomial Division
Polynomial Long Division
Synthetic Division
• Always works
• Divisor MUST be in
the form (x – r)
– Use the normal
algorithm for long
division
– x cannot be raised to
any power other
than one to use
synthetic division!
Polynomial Long Division
polynomial
remainder
f  x
d  x
 q  x 
r  x
d  x
divisor
divisor
quotient
1. Make sure both the polynomial and divisor are in standard form.
(decreasing order of degree)
2. If terms are missing, put them in with 0 coefficients.
3. The polynomial goes inside the house and the divisor goes outside. (like
regular long division)
4. Focus on the first term and what you’d have to multiply the first term of the
divisor by to get the first term of the polynomial.
5. Continue multiplying and subtracting like regular long division until the
remainder is one degree less than the divisor.
Glencoe – Algebra 2
Chapter 5: Polynomials
8
Example 1
62
8 497
Divide 497 by 8.
48
17
16
1
497
1
 62 
8
8
Glencoe – Algebra 2
Chapter 5: Polynomials
9
Simplify:


2 x 3  11x 2  10 x  6   x  4 
Rewrite as follows:
2
2x 3x 2 
x4
3
2
x  4 2 x  11x  10 x  6
 3  2
2x  8 x
2
Change
the
signs.
3 x  10 x
Change
Change
the
the
signs.
signs.
Scrap
Paper
Ask
yourself…x
times what gives 2x3?
  3 x 2  12 x
Scrap
Paper
Scrap
Paper
2
2ask
x xyourself…x
x

4


2times

2
x
Now
Answer…x
times
2x
gives what
2x3. gives
3x2? 6
3
x

4


2  x  4 
 

2
Answer…x3 times 3x
2 x  8
2 gives 3x .
23xx 2812
xx
2
 2 x  8
2
Now ask yourself…x times what gives -2x?
Answer…x times -2 gives -2x.
Your exponents must go in descending order.
If you are missing an exponent, put in a zero for
that place.
Example:
x 3  1 should be written as: x 3  0 x 2  0 x  1
You must change signs before you add!!!
Write remainders as fractions.
One last example
Divide 2x 4  3x 3  1  5x
by 2  x 2  2x .
2x 2  7x  10
x 2  2x  2 2x 4  3x 3  5
0x 21 5x  1

2x 4  4x
4x 3  4x
4x 2
7x3  4x 2  5x
77x
x 33 14x
14x 22 14x
14x
10x2  9x  1

10x 22 20x
20x  20
11x  21
11x  21
2x 4  3x 3  5x  1
2

2
x

7
x

10

x 2  2x  2
x 2  2x  2
Glencoe – Algebra 2
Chapter 5: Polynomials
12
Try these:
x  3x  5  x  1
2
x 1  x 1
3
2 x  4 x  5 x  3x  2  x  2 x  3
4
3
2
2
Getting the problem set up.
3 x 4  7 x 3  x  11   x  3 


First, make sure there are no skipped powers.
Rewrite with zeros if necessary.
3x
4

 7 x  0x  x  11   x  3 
3
2
Next, write just the coefficients of the dividend.
-3 3
7
0
1
-11
Then, find out what value makes the divisor equal
Finally, skip a line and draw a line.
zero and write that number in the “box”.
Getting’ it done.
-3 3
3
7
-9
-2
0
1
6 -18
6 -17
-11
51
40
1. Bring down the first number.
2. Multiply the number in the “box” by this number.
3. Place your answer under the next number.
4. Add.
5. Repeat 2-4.
Now what?!?
-3 3
3
7
-9
-2
0
6
6
1
-18
-17
-11
51
40
40
3 x  2 x  6 x  17 
x 3
3
2
Box your last number. This is your remainder.
Your first variable’s exponent will be one less than
the dividend’s. The remaining exponents go in
descending order.
Example 3
Divide 2x 3  x 2  8x  5
by x  3.
3 2 1 8 5
6 15 21
5 7 16
2x 3  x 2  8x  5
x 3
Glencoe – Algebra 2
Chapter 5: Polynomials
 2x 2  5x  7 
16
x 3
18
Example 4
Factor 3x  4x  28x  16
3
2
completely given that x  2
is a factor.
2 3 4 28 16
6 20 16
10 8
0

2
3x 3  4x 2  28x  16   x  2  3x  10x  8

  x  2 3x  2  x  4 
Glencoe – Algebra 2
Chapter 5: Polynomials
19
Try these
• Divide using synthetic division
x  10 x  2 x  4  x  3
4
2
2 x  7 x  4 x  27 x  18  x  2
4
3
2
Synthetic Division
Class/Homework
• Handout
• Oh yes, and one more thing (next slide…)
Dividing a Polynomial by a Monomial
 Divide each term of the polynomial by the
monomial.
 Remember to divide coefficients and subtract
exponents.
Divide 12x2 – 20x + 8 by 4x
12x2 – 20x + 8
4x
4x
3x  5
4x
2

x
Examples
1.
Divide 9x2 + 12x – 18 by 3x.
6
3x  4 
x
2.
(Do in your notes) Divide 32x2 – 16x + 64 by -8x
8
 4x  2 
x
Practice
1. Divide 18x2 + 45x – 36 by 9x
Answers:
4
1.2 x  5 
x
2. Divide 10b3 – 8b2 -5b by – 2b
5
2. 5b  4b 
2
3. Divide x2 – 8x + 15 by x – 3
3. x  5
4. Divide 5x2 + 3x – 15 by x + 2
1
4. 5 x  7 
x2
2