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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)

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Which of the following polynomials has a double root? a)x 2 -5x+6=(x-2)(x-3) b)x 2 -4x+4=(x-2)(x-2) c)x 4 -14x 2 +45=(x 2 -9)(x 2 -5)=(x-3)(x+3)(x-√5)(x+√5) d)Both (a) and (b) e)Both (b) and (c) B

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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) B

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Polynomial Division

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Factoring Polynomials Let’s say I have a polynomial x 3 -6x 2 +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 x 3 -6x 2 +32=(x-4)(…………….)

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Polynomial Division x 3 -6x 2 +32=(x-4)(…………….) In order to find the (…………….), I have to divide both sides by (x-4). (x 3 -6x 2 +32)/(x-4)=(…………….) Now I need a way to divide polynomials.

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Two Methods Polynomial Long Division – Long, takes up a lot of space – Easier to read Synthetic Division – Short, fast

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Polynomial Long Division

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(x 3 -6x 2 +32)/(x-4) Write out the factor, the division sign, and the full polynomial x-4 |x 3 -6x 2 +0x+32

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(x 3 -6x 2 +32)/(x-4) x 3 /x =x 2, put x 2 on top x 2 x-4 |x 3 -6x 2 +0x+32

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(x 3 -6x 2 +32)/(x-4) x 2 (x-4)=x 3 -4x 2, put x 3 -4x 2 underneath an line it up. x 2 x-4 |x 3 -6x 2 +0x+32 x 3 -4x 2

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(x 3 -6x 2 +32)/(x-4) Subtract down to get a new polynomial x 2 x-4 |x 3 -6x 2 +0x+32 x 3 -4x 2 -2x 2 +0x+32

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(x 3 -6x 2 +32)/(x-4) Repeat steps: divide to the top (-2x 2 /x), multiply to the bottom (-2x(x-4)), subtract down. x 2 -2x x-4 |x 3 -6x 2 +0x+32 x 3 -4x 2 -2x 2 +0x+32 -2x 2 +8x -8x+32

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(x 3 -6x 2 +32)/(x-4) Repeat steps: divide to the top (-8x/x), multiply to the bottom (-8(x-4)), subtract down. x 2 -2x -8 x-4 |x 3 -6x 2 +0x+32 x 3 -4x 2 -2x 2 +0x+32 -2x 2 +8x -8x+32 0

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(x 3 -6x 2 +32)/(x-4) Our remainder is 0, meaning that x-4 really is a factor of x3-6x2+32 x 2 -2x -8 x-4 |x 3 -6x 2 +0x+32 x 3 -4x 2 -2x 2 +0x+32 -2x 2 +8x -8x+32 0

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(x 3 -6x 2 +32)/(x-4) Write down the factorization x 3 -6x 2 +0x+32=(x-4)(x 2 -2x-8) x 2 -2x -8 x-4 |x 3 -6x 2 +0x+32 x 3 -4x 2 -2x 2 +0x+32 -2x 2 +8x -8x+32 0

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Example with a remainder x 2 -2x -8 x-4 |x 3 -6x 2 x 3 -4x 2 -2x 2 -2x 2 +8x -8x -8x+32 -32

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Example with a remainder x 2 -2x -8 x-4 |x 3 -6x 2 x 3 -4x 2 -2x 2 -2x 2 +8x -8x -8x+32 -32 x-4 Is NOT a factor of x 3 -6x 2

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Example with a remainder x 2 -2x -8 x-4 |x 3 -6x 2 x 3 -4x 2 -2x 2 -2x 2 +8x -8x -8x+32 -32 x-4 Is NOT a factor of x 3 -6x 2

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Synthetic Division

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Does exactly the same thing as polynomial long division – Faster – Takes up less space – Easier (for me, at least)

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Factoring Polynomials Let’s say I have a polynomial x 3 -6x 2 +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 x 3 -6x 2 +32=(x-4)(…………….)

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Polynomial Division x 3 -6x 2 +32=(x-4)(…………….) In order to find the (…………….), I have to divide both sides by (x-4). (x 3 -6x 2 +32)/(x-4)=(…………….)

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What is the quotient when the polynomial 3x 3 − 18x 2 − 27x + 162 is divided by x-3? a)3x 2 +9x-54 b)3x 2 +9x+54 c)3x 2 -9x+54 d)3x 2 -9x-54 e)None of the above is completely correct

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What is the quotient when the polynomial 3x 3 − 18x 2 − 27x + 162 is divided by x-3? 3x 2 -9x -54 x-3 |3x 3 -18x 2 -27x+162 3x 3 -9x 2 -9x 2 -27x+162 -9x 2 +27x -54x+162 0 3 -9 -54 0 3 |3 -18 -27 162 9 -27 -162 D) 3x 2 -9x-54

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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)=3x 3 -18x 2 -27x+162 3 -9 -54 0 3 |3 -18 -27 162 9 -27 -162 Remainder is 0, so ƒ(3)=0

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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)=3x 3 -18x 2 -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 3 is a root (x-3) is a factor

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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)=3x 3 -18x 2 -27x+162 3 -15 -42 120 1 |3 -18 -27 162 3 -15 -42 Remainder is 120, so ƒ(1)=120

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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)=3x 3 -18x 2 -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 1 is NOT a root (x-1) is NOT a factor

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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 the same as

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Which of the following is a linear factor of f(x) = x 3 - 6x 2 + 21x - 26? a) x - 2 b) x + 2 c) x d) (a) and (b) e) None of the above

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Finding Roots With the Remainder Theorem. Dividing Polynomials Remember, when we divide a polynomial by another polynomial, we get a quotient and a remainder.

Finding Roots With the Remainder Theorem. Dividing Polynomials Remember, when we divide a polynomial by another polynomial, we get a quotient and a remainder.

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