advanced algebra notes section 5.5: apply the remainder and factor theorems common core standard:...
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Advanced Algebra Notes
Section 5.5: Apply the Remainder and Factor Theorems
Common Core Standard: A-APR #2Know and apply the Remainder Theorem
Advanced Algebra NotesSection 5.5: Apply the Remainder and Factor Theorems
Remember when you used to use long division when solving a basic general math problem like:
5 436
8
40
36
7
35
1
1
5
436 5
Definitions from the problem above:
1. Dividend:
2. Divisor:
3. Quotient:
4. Remainder:
Number inside the division box (436)
Number outside the division box (5)
Number above the division box ( )
1875
Number that gets written as the numerator of the fraction (1)
We are going to do something very similar to that but with polynomials, and it is called _________________.long division
Examples: Divide these polynomials using long division.
)6()7132( 2 xxx
1.
26 2 13 7x x x 2x
22 12x x
7x
1
6x
13
13
6x
2)
4 2 2( 2 5) ( 1)y y y y y
2 4 3 21 2 5y y y oy y y
2y
4 3 2y y y
3 2 5y y y
y
3 2y y y
22 2 5y y
2
22 2 2y y
3
2
3
1y y
We can use another method to divide polynomials as long as the divisor is in the form x – k. This method is called _________________, which is exactly the same method as synthetic substitution that we did in section 5.2.
synthetic division
Remainder Theorem: If a polynomial f(x) is divided by x – k, then the remainder is r = f(k).
Example: Divide using synthetic division.
3) Divide f(x) = 2x3 + 9x2 + 14x + 5 by x – 3
3 2 9 14 5
2
6
15
45
59
177
182
3
18259152 2
x
xx
4) Divide f(x) = -4x3 + 5x2 + 8 by x + 3
3 8054
4
12
17
51
51
153
161
3
16151174 2
x
xx
Suppose the remainder is 0 when a polynomial f(x) is divided by x – k: so x – k is a __________of the dividend f(x). Factor Theorem: A polynomial f(x) has a factor x – k if and only if f(k) = 0. The factor theorem can be used to solve a variety of problems. Problem Example Given one factor of a polynomial, find the other factors. Example 5 belowGiven one zero of a polynomial function, find the other zeros. Example 6
Examples: Factor completely.
5) f(x) = 2x3 – 11x2 + 3x + 36 given that x – 3 is a factor
factor
3 363112
2
6
5
-15
-12
-36
0
)1252)(3( 2 xxx
)4)(32)(3( xxx
Fix this
6) One zero of f(x) = x3 + x2 – 16x – 16 is 4. What is another zero of f(x)?
4 161611
1
4
520
4
16
0
0452 xx
0)1)(4( xx
0104 xx
14 xx
A company’s profit C (in thousands of dollars) can be modeled by C = -5x3 + 6x2 + 15x, where x is the number of items produced in thousands. The profit is $14,000 for producing 2000 items. What other number of items would produce about the same profit?