Embed Size (px)
DESCRIPTION
Remainder Theorem. If the polynomial f(x) is divided by x c, then the remainder is equal to f(c). Example 1. In other words, the remainder after performing synthetic division is the same number we would get if we replaced c into the polynomial and evaluated the polynomial. - PowerPoint PPT Presentation
Citation preview
1
2.2 Remainder and Factor Theorems
Remainder Theorem
If the polynomial f(x) is divided by x c, then the remainder is equal to f(c).
In other words, the remainder after performing synthetic division is the same number we would get if we replaced c into the polynomial and evaluated the polynomial.
Example 1. 3 2f(x) 5x 3x 21x 1
a) Find f(2) by evaluating f(2) directly. b) Find f(2) using synthetic division and the remainder theorem.3 2f( 2) 5( 2) 3( 2) 21( 2) 1
5( 8) 3(4) 21( 2) 1 40 12 42 1
f( 2) 13
Using synthetic division, let c = -2.
5 3 21 12
5
107 7
14 14
13
Your Turn Problem #13 2If f(x) = 3x 5x 3x 10, find f(-2) by
a) evaluating f (-2) directly and
b) Use synthetic division and the remainder theorem.
Both methods give a result of 13.
Answer: f( 2) 60
2
2.2 Remainder and Factor Theorems
5 fExa (n)mple 2. 2n 3
a) Find f(2) by evaluating f(2) directly.b) Find f(2) using synthetic division and the remainder theorem.
5f (2) 2(2) 3 2(32) 3 64 3
f (2) 61
Using synthetic division, let c = 2.
2 0 0 0 0 32
2 61
4
16 32
16 32 64
4
8
8
Both methods give a result of 61.
Your Turn Problem #24If f(x) = 2x 7, find f(-3) by
a) evaluating f (-3) directly and
b) Use synthetic division and the remainder theorem.
Answer: f( 3) 155
3
2.2 Remainder and Factor Theorems
Factor TheoremA polynomial f(x) has a factor x c if and only if f(c) = 0.
In other words, if the remainder after performing synthetic division is zero or the result from evaluating the polynomial at c is zero, then x c is a factor.
We can use two different methods to answer the question. Either evaluate f(c) directly or use synthetic division. We’ll first evaluate f(c) directly.
3 2 a) Is x 3 a factor ofExa 3x 14xmple 173. x 6?
We are given x 3. Therefore use c = 3.
3 2f (3) 3(3) 14(3) 17(3) 6
f (3) 0
3(27) 14(9) 17(3) 6 81 126 51 6 Answer: Yes. If f(c) = 0, then the
divisor is a factor of the polynomial
3 2 b) Is x 3 a factor of xExamp 4le 3. x 9x 36?
The other method we can use is synthetic division. Often this method is more preferable because we can obtain more information than just the remainder.
We are given x + 3. Therefore use c = 3.
Show by using synthetic division.
Next Slide
Answer: Yes, f(3) = 0.
1 4 9 363
1
37 12
21 360
4
2.2 Remainder and Factor Theorems
Your Turn Problem #33 2a) Is x 3 a factor of 2x 4x 29x 3?
Show by evaluating f(c) directly.
3 2b) Is x 2 a factor of x 4x 4x 16?
Show using synthetic division.
Answer: No, f(-3) =-6.
2 1 4 4 16
1
2
2 84 16
0Answer: Yes, since f(2)=0.
5
2.2 Remainder and Factor Theorems
Let’s look further at the last your turn problem.
2 1 4 4 16
1
2
2 84 16
0
3 2x 2 is a factor of x 4x 4x 16 because
Recall from the previous section, the bottom row gives us the quotient. 2The quotient is x 2x 8.
3 2 2Therfore x 4x 4x 16 = (x 2)(x 2x 8)
2Also notice that x 2x 8 can be factored as (x 4)(x 2).
We can then write the polynomial completely factored as: (x 2)(x 4)(x 2)
Procedure: To factor a polynomial P(x) given a factor x c.
1. Use synthetic division to show x c is a factor of P(x) by showing remainder = 0.2. Rewrite the quotient in proper form with variables.
3. Factor the quotient (if possible) using previous factoring techniques.
Next Slide
4. The polynomial P(x) completely factored = (x c)(quotient factored)
6
2.2 Remainder and Factor Theorems
1st, show g(x) is a factor of f(x).
2nd, rewrite the quotient in proper form.
2 1 6 13 42
1
2
4 218 42
0
Show the complete factorization.
Example 4. Show g(x) is a factor of f(x) and complete the factorization of f(x). 3 2g(x)= x 2, f(x) x 6x 13x 42
Solution:
2x 4x 21
3rd, factor the quotient.
(x 7)(x 3)
Answer: (x 2)(x 7)(x 3)
Your Turn Problem #4
Show g(x) is a factor of f(x) and complete the factorization of f(x).
3 2g(x) = x 2 and f(x) = x 3x 10x 24
Answer: f(x) = (x 3)(x 2)(x 4)
7
2.2 Remainder and Factor Theorems
1st, show g(x) is a factor of f(x).
2nd, rewrite the quotient in proper form.
2 6 5 12 4
614
7 212 4
0Show the complete factorization.
Example 5.
3 2g(x)= x 2, f(x) 6x 5x 12x 4
Show g(x) is a factor of f(x) and complete the factorization of f(x).
Solution:
26x 7x 2
3rd, factor the quotient.
(2x 1)(3x 2)
Answer: (x 2)(2x 1)(3x 2)
Your Turn Problem #5
Show g(x) is a factor of f(x) and complete the factorization of f(x).
3 2g(x) = x 4 and f(x) = 3x 13x 6x 40
Answer: f(x) = (x 4)(3x 5)(x 2)
8
2.2 Remainder and Factor Theorems
Example 6.
5 4g(x)= x 1, f(x) x x 16x 16
Show g(x) is a factor of f(x) and complete the factorization of f(x).
Solution:
2 2x 4 x 4
2Answer: x+1 x 2 x 2 x 4
4x 16 1 1 1 0 0 16 16
1 0
1
0 16
0 0 16
0
0
0 2x 2 x 2 x 4
Your Turn Problem #6
Show g(x) is a factor of f(x) and complete the factorization of f(x).
5 4g(x) = x 1 and f(x) = x x 16x 16
2Answer: f(x) = x 1 x 2 x 2 x 4
The End.B.R.3-5-07
