Download 2.5 Divide Polynomials revised.notebook

2.5 Divide Polynomials revised.notebook
November 02, 2016
Warm Up
Solve each polynomial equation by factoring.
1. x6 ­ 8x3 = 0
2. x4 ­ x3 ­ 20x2 = 0
3. x4 + 12x3 + 4x2 = 0
4. x5 ­ 256x = 0
I. Long Division of Polynomials
Let's review long division.
quotient
divisor
dividend
2.5 Divide Polynomials revised.notebook
November 02, 2016
Let's apply long division to polynomials.
STEPS:
1. Write the dividend in descending order
with every power of the variable
represented.
2. Divide the term with the highest power (in the
dividend) by the leading term of the divisor.
Write the result as the next term of the quotient.
3. Multiply the term of the quotient with the divisor.
Write the product below the dividend.
4. Subtract the product of step 3 by adding the
opposite of each term.
5. Bring down the next term of the dividend.
6. Repeat steps 2-5 until remainder is a lower
degree than the divisor.
CHECK YOUR ANSWER: Multiply the divisor and
quotient, then add the remainder.
2.5 Apply the Remainder and Factor Theorems
Polynomial Division
remainder
polynomial
divisor
quotient
When you divide a polynomial by a divsior, you get a quotient and a remainder.
The degree of the remainder must be less than the degree of the divisor.
2.5 Divide Polynomials revised.notebook
November 02, 2016
EXAMPLE 2
STEPS:
1. Write the dividend in descending order
with every power of the variable
represented.
2. Divide the term with the highest power (in the
dividend) by the leading term of the divisor.
Write the result as the next term of the quotient.
3. Multiply the term of the quotient with the divisor.
Write the product below the dividend.
4. Subtract the product of step 3 by adding the
opposite of each term.
5. Bring down the next term of the dividend.
6. Repeat steps 2-5 until remainder is a lower
degree than the divisor.
CHECK YOUR ANSWER: Multiply the divisor and
quotient, then add the remainder.
EXAMPLE 3
2.5 Divide Polynomials revised.notebook
November 02, 2016
TRY THESE:
4
2
2
1. (x + 2x - x + 5)
3
2
(x - x + 1)
2. (x - x - 2x + 8)
3
2
3. (x - 6x + 1)
(x + 2)
(x - 2)
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)
2.5 Divide Polynomials revised.notebook
November 02, 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.
3
2
2. (x - x - 2x + 8)
(x + 2)
2.5 Divide Polynomials revised.notebook
November 02, 2016
Divide by synthetic division.
3
2
3. (x - 6x + 1)
(x - 2)
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?
2.5 Divide Polynomials revised.notebook
November 02, 2016
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
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
2.5 Divide Polynomials revised.notebook
November 02, 2016
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.
3
2
2. Find all zeros of 3x - 11x - 6x + 8, given that f(4) = 0.
2.5 Divide Polynomials revised.notebook
November 02, 2016
The factor theorem can be used if you are
given one factor.
3
2
3. Completely factor f(x) = 2x - 9x - 32x - 21 given that x - 7
is a factor.
The factor theorem can be used if you are
given one solution.
3
2
4. Find all solutions to x - 11x + 28x - 90 = 0 given that x = 9 is
one solution.
2.5 Divide Polynomials revised.notebook
November 02, 2016
Additional Practice
Find the remaining zeros, given one zero.
1. 2x4 + 3x3 ­ 11x2 ­ 9x + 5 ; x = 1
2. x3 + 6x2 + 21x + 26; x = ­2
3. 8x3 ­ 6x2 ­ 23x + 6 ; x = 2
4. 3x3 + 14x2 ­ 28x ­ 24 ; x = ­2/3
5. x3 ­ 2x2 ­ 9x + 18 ; x = ­3