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Which of the following polynomials has a double root?

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

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Which of the following polynomials has a double root?

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  1. Which of the following polynomials has a double root? • x2-5x+6 • x2-4x+4 • x4-14x2+45 • Both (a) and (b) • Both (b) and (c)

  2. Which of the following polynomials has a double root? • x2-5x+6 =(x-2)(x-3) • x2-4x+4 =(x-2)(x-2) • x4-14x2+45 =(x2-9)(x2-5)=(x-3)(x+3)(x-√5)(x+√5) • Both (a) and (b) • Both (b) and (c) B

  3. Which of the following polynomials has a double root? • x2-5x+6 • x2-4x+4 • x4-14x2+45 • Both (a) and (b) • Both (b) and (c) B

  4. Polynomial Division

  5. 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)(…………….)

  6. 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.

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

  8. Polynomial Long Division

  9. (x3-6x2+32)/(x-4) • Write out the factor, the division sign, and the full polynomial x-4 |x3-6x2+0x+32

  10. (x3-6x2+32)/(x-4) • x3/x =x2, put x2 on top x2 x-4 |x3-6x2+0x+32

  11. (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

  12. (x3-6x2+32)/(x-4) • Subtract down to get a new polynomial x2 x-4 |x3-6x2+0x+32 x3-4x2 -2x2+0x+32

  13. (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

  14. (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

  15. (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

  16. (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

  17. Example with a remainder x2-2x -8 x-4 |x3-6x2 x3-4x2 -2x2 -2x2+8x -8x -8x+32 -32

  18. 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

  19. 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

  20. Synthetic Division

  21. Synthetic Division • Does exactly the same thing as polynomial long division • Faster • Takes up less space • Easier (for me, at least)

  22. 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)(…………….)

  23. 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)=(…………….)

  24. What is the quotient when the polynomial 3x3 − 18x2 − 27x + 162 is divided by x-3? • 3x2+9x-54   • 3x2+9x+54 • 3x2-9x+54    • 3x2-9x-54  • None of the above is completely correct

  25. 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

  26. 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

  27. 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

  28. 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

  29. 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

  30. 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| 3 -18 -27 162 9 -27 -162 3 -9 -54| 0 3-9 -54 0 3|3 -18 -27 162 9 -27 -162 Is the same as

  31. Which of the following is a linear factor of f(x) = x3 - 6x2 + 21x - 26? a) x - 2 b) x + 2 c) x d) (a) and (b) e) None of the above

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