Download 10-18 NOTES Divide Polynomials and Find Remaining Zeros

2.5 Divide Polynomials.notebook
October 18, 2016
Warm Up
1. Solve by factoring.
x5 ­ 256x = 0
2. Divide.
(2x4 ­ 6) ÷ (x + 2)
Warm Up
More Practice
Divide using long division.
4
3
1. (x - 3x + 5x - 6) (x + 2)
2
2. (6x - x - 7)
3
3. (2x + 4x - 6)
3
(3x + 1)
(x + 3)
2
4. (4x - 8x + 3x - 8)
(2x - 1)
Oct 27­2:38 PM
1
2.5 Divide Polynomials.notebook
October 18, 2016
Answers: Long Division
1. x3 ­ 5x2 + 10x ­ 15 R 24
2. 2x ­ 1 R ­6
3. 2x2 ­ 6x + 22 R ­72
4. 2x2 ­ 3x R ­8
HW Answers Long Div
Review
Use synthetic substitution to evaluate
f(x) = x4 ­ 2x3 + x2 ­ 3x + 2 for x = 2.
Connect
4
3
2
Divide using long division. (x - 2x + x - 3x + 2)
(x - 2)
Synthetic Sub. & Long Div
2
2.5 Divide Polynomials.notebook
October 18, 2016
II. Synthetic Division of Polynomials (shortened method of long division)
Divisor MUST HAVE the form (x - k).
4
3
1. (x + 4x
+ 16x - 35)
(x + 5)
STEPS:
1. Write the dividend in descending order
with every power of the variable
represented.
2. Write the coefficients of the dividend
in a row (don't write the variables).
3. Write the opposite of the constant
from the divisor in front of the row
of coefficients.
4. Bring first coefficient down below the
line.
5. Multiply the number with divisor in
front and write product under the
next coefficient.
6. Add numbers in the column and write
the sum below the line.
7. Repeat steps 5 and 6 until finished.
Synthetic Division
3
2
2. (x - x - 2x + 8)
(x + 2)
EX2: Syn Div
3
2.5 Divide Polynomials.notebook
October 18, 2016
III. Remainder Theorem
If a polynomial f(x) is divided by (x - k), the remainder is
r = f(k).
3
2
(x - 2)
(x - 6x + 1)
EXAMPLE:
5
3
1. What is the remainder when f(x) = 3x - 5x + 57 is divided by
(x - 2)?
4
3
2. What is the remainder when f(x) = x - 2x + x - 1 is divided by
(x + 1)?
4
3
3. Is x = -2 a zero of f(x) = x + 2x - 8x - 16?
Remainder Theorem
IV. Factor Theorem
A polynomial has a factor (x - k) if and only if f(k) = 0.
EXAMPLES:
Determine if the following polynomials are factors of
3
2
f(x) = x + 6x - x - 30.
1. x - 2
2. x - 3
3. x + 5
Factor Theorem
4
2.5 Divide Polynomials.notebook
October 18, 2016
IV. Zeros of Polynomials
If (x ­ k) is a factor of a polynomial f(x), then f(k) = 0, AND k is a zero of f(x).
If k is a zero of f(x), then f(k) = 0 AND (x ­ k) is a factor of the polynomial f(x).
If the number k is a zero of a polynomial function f(x), then all of the
following are true:
1.
2.
3.
4.
5.
k is a solution, or root, of the polynomial equation f(x) = 0.
(x ‐ k) is a factor of the polynomial f(x).
f(k) = 0.
if the polynomial f(x) is divided by (x ‐ k), the remainder is 0.
if a real number,k is an x‐intercept on the graph of the polynomial
function f(x).
Example:
y = 2x3 ­ 4x2 ­ 6x
Zeros of Polys
The factor theorem can be used if you are
given one zero.
3
2
1. Find the remaining zeros of 2x + 11x + 18x + 9, if f(-3) = 0.
Find remaining zeros
5
2.5 Divide Polynomials.notebook
October 18, 2016
3
2
2. Find all zeros of 3x - 11x - 6x + 8, given that f(4) = 0.
Find remaining zeros
Homework
Find the remaining zeros, given one zero.
1. x3 ­ 4x2 ­ 7x + 10 ; x = 1
2. 2x3 ­ 3x2 ­ 11x + 6; x = ­2
3. x3 ­ 3x2 ­ 3x + 9 ; x = 3
4. 2x4 + 3x3 ­ 6x2 ­ 6x + 4; x = 1/2
5. x3 ­ 3x2 + 9x + 13 ; x = ­1
6. x3 + 3x2 ­ 2x ­ 6 ; x = ­3
Oct 27­2:38 PM
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