Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45...
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Transcript of Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45...
![Page 1: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/1.jpg)
Which of the following polynomials has a double root?
a) x2-5x+6
b) x2-4x+4
c) x4-14x2+45
d) Both (a) and (b)
e) Both (b) and (c)
![Page 2: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/2.jpg)
Which of the following polynomials has a double root?
a) x2-5x+6 =(x-2)(x-3)
b) x2-4x+4 =(x-2)(x-2)
c) x4-14x2+45 =(x2-9)(x2-5)=(x-3)(x+3)(x-√5)(x+√5)
d) Both (a) and (b)
e) Both (b) and (c)
B
![Page 3: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/3.jpg)
Which of the following polynomials has a double root?
a) x2-5x+6
b) x2-4x+4
c) x4-14x2+45
d) Both (a) and (b)
e) Both (b) and (c)
B
![Page 4: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/4.jpg)
Polynomial Division
![Page 5: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/5.jpg)
Factoring Polynomials
• Let’s say I have a polynomial x3-6x2+32 and I want to factor it.– Factoring cubics is hard.
• Maybe I graph it and I notice that it looks like I have a root at x=4.– I can guess that my factoring will look something
like• x3-6x2+32=(x-4)(…………….)
![Page 6: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/6.jpg)
Polynomial Division
• x3-6x2+32=(x-4)(…………….)• In order to find the (…………….), I have to divide
both sides by (x-4).• (x3-6x2+32)/(x-4)=(…………….)• Now I need a way to divide polynomials.
![Page 7: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/7.jpg)
Two Methods
• Polynomial Long Division– Long, takes up a lot of space– Easier to read
• Synthetic Division– Short, fast
![Page 8: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/8.jpg)
Polynomial Long Division
![Page 9: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/9.jpg)
(x3-6x2+32)/(x-4)
• Write out the factor, the division sign, and the full polynomial
x-4 |x3-6x2+0x+32
![Page 10: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/10.jpg)
(x3-6x2+32)/(x-4)
• x3/x =x2, put x2 on top
x2 x-4 |x3-6x2+0x+32
![Page 11: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/11.jpg)
(x3-6x2+32)/(x-4)
• x2(x-4)=x3-4x2, put x3-4x2 underneath an line it up.
x2 x-4 |x3-6x2+0x+32
x3-4x2
![Page 12: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/12.jpg)
(x3-6x2+32)/(x-4)
• Subtract down to get a new polynomial
x2 x-4 |x3-6x2+0x+32
x3-4x2
-2x2+0x+32
![Page 13: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/13.jpg)
(x3-6x2+32)/(x-4)
• Repeat steps: divide to the top (-2x2/x), multiply to the bottom (-2x(x-4)), subtract down.
x2-2x x-4 |x3-6x2+0x+32
x3-4x2
-2x2+0x+32 -2x2+8x
-8x+32
![Page 14: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/14.jpg)
(x3-6x2+32)/(x-4)
• Repeat steps: divide to the top (-8x/x), multiply to the bottom (-8(x-4)), subtract down.
x2-2x -8 x-4 |x3-6x2+0x+32
x3-4x2
-2x2+0x+32 -2x2+8x
-8x+32 -8x+32
0
![Page 15: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/15.jpg)
(x3-6x2+32)/(x-4)
• Our remainder is 0, meaning that x-4 really is a factor of x3-6x2+32
x2-2x -8 x-4 |x3-6x2+0x+32
x3-4x2
-2x2+0x+32 -2x2+8x
-8x+32 -8x+32
0
![Page 16: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/16.jpg)
(x3-6x2+32)/(x-4)
• Write down the factorization x3-6x2+0x+32=(x-4)(x2-2x-8)
x2-2x -8 x-4 |x3-6x2+0x+32
x3-4x2
-2x2+0x+32 -2x2+8x
-8x+32 -8x+32
0
![Page 17: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/17.jpg)
Example with a remainder
x2-2x -8 x-4 |x3-6x2
x3-4x2
-2x2
-2x2+8x -8x -8x+32
-32
![Page 18: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/18.jpg)
Example with a remainder
x2-2x -8 x-4 |x3-6x2
x3-4x2
-2x2
-2x2+8x -8x -8x+32
-32 x-4 Is NOT a factor of x3-6x2
![Page 19: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/19.jpg)
Example with a remainder
x2-2x -8 x-4 |x3-6x2
x3-4x2
-2x2
-2x2+8x -8x -8x+32
-32 x-4 Is NOT a factor of x3-6x2
![Page 20: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/20.jpg)
Synthetic Division
![Page 21: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/21.jpg)
Synthetic Division
• Does exactly the same thing as polynomial long division– Faster– Takes up less space– Easier (for me, at least)
![Page 22: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/22.jpg)
Factoring Polynomials
• Let’s say I have a polynomial x3-6x2+32 and I want to factor it.– Factoring cubics is hard.
• Maybe I graph it and I notice that it looks like I have a root at x=4.– I can guess that my factoring will look something
like• x3-6x2+32=(x-4)(…………….)
![Page 23: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/23.jpg)
Polynomial Division
• x3-6x2+32=(x-4)(…………….)• In order to find the (…………….), I have to divide
both sides by (x-4).• (x3-6x2+32)/(x-4)=(…………….)
![Page 24: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/24.jpg)
![Page 25: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/25.jpg)
![Page 26: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/26.jpg)
![Page 27: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/27.jpg)
![Page 28: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/28.jpg)
![Page 29: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/29.jpg)
![Page 30: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/30.jpg)
![Page 31: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/31.jpg)
![Page 32: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/32.jpg)
What is the quotient when the polynomial 3x3 − 18x2 − 27x + 162 is divided by x-3?
a) 3x2+9x-54
b) 3x2+9x+54
c) 3x2-9x+54
d) 3x2-9x-54
e) None of the above is completely
correct
![Page 33: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/33.jpg)
What is the quotient when the polynomial 3x3 − 18x2 − 27x + 162 is divided by x-3?
3x2 -9x -54 x-3 |3x3-18x2-27x+162 3x3-9x2
-9x2-27x+162 -9x2+27x
-54x+162-54x+162
0
3 -9 -54 0 3 |3 -18 -27 162 9 -27 -162
D) 3x2-9x-54
![Page 34: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/34.jpg)
Fun Tricks with Synthetic Division
• If you divide ƒ(x) and (x-c), then the remainder is the value of ƒ(c)
Example: ƒ(x)=3x3-18x2-27x+162
3 -9 -54 0 3 |3 -18 -27 162 9 -27 -162
Remainder is 0, so ƒ(3)=0
![Page 35: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/35.jpg)
Fun Tricks with Synthetic Division
• If you divide ƒ(x) and (x-c), then the remainder is the value of ƒ(c)
Example: ƒ(x)=3x3-18x2-27x+162
3 -9 -54 0 3 |3 -18 -27 162 9 -27 -162
Remainder is 0, so ƒ(3)=0 3 is a root
(x-3) is a factor
![Page 36: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/36.jpg)
Fun Tricks with Synthetic Division
• If you divide ƒ(x) and (x-c), then the remainder is the value of ƒ(c)
Example: ƒ(x)=3x3-18x2-27x+162
3 -15 -42 120 1 |3 -18 -27 162 3 -15 -42
Remainder is 120, so ƒ(1)=120
![Page 37: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/37.jpg)
Fun Tricks with Synthetic Division
• If you divide ƒ(x) and (x-c), then the remainder is the value of ƒ(c)
Example: ƒ(x)=3x3-18x2-27x+162
3 -15 -42 120 1 |3 -18 -27 162 3 -15 -42
Remainder is 120, so ƒ(1)=120 1 is NOT a root
(x-1) is NOT a factor
![Page 38: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/38.jpg)
Final Thought
• Your book (and possibly your recitation instructor) write synthetic division upside down. It’s the same thing, just with the numbers in a different place.
3 -9 -54 0 3 |3 -18 -27 162 9 -27 -162
3 | 3 -18 -27 162 9 -27 -162 3 -9 -54 | 0
Is thesame as
![Page 39: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/39.jpg)
Which of the following is a linear factor of f(x) = x3 - 6x2 + 21x - 26?
a) x - 2b) x + 2 c) x d) (a) and (b)e) None of the above
![Page 40: Which of the following polynomials has a double root? a)x 2 -5x+6 b)x 2 -4x+4 c)x 4 -14x 2 +45 d)Both (a) and (b) e)Both (b) and (c)](https://reader030.fdocuments.in/reader030/viewer/2022020111/56649c7c5503460f94930d8b/html5/thumbnails/40.jpg)
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