which of the following polynomials has a double root?
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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) - PowerPoint PPT PresentationTRANSCRIPT
Find the inverse of the function f(x)=3x-5.
Which of the following polynomials has a double root?
x2-5x+6x2-4x+4x4-14x2+45Both (a) and (b)Both (b) and (c)
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
Which of the following polynomials has a double root?
x2-5x+6x2-4x+4x4-14x2+45Both (a) and (b)Both (b) and (c)
BPolynomial DivisionFactoring PolynomialsLets 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 likex3-6x2+32=(x-4)(.)Polynomial Divisionx3-6x2+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.Two MethodsPolynomial Long DivisionLong, takes up a lot of spaceEasier to readSynthetic DivisionShort, fastPolynomial Long Division(x3-6x2+32)/(x-4)Write out the factor, the division sign, and the full polynomial
x-4 |x3-6x2+0x+32(x3-6x2+32)/(x-4)x3/x =x2, put x2 on top
x2 x-4 |x3-6x2+0x+32(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(x3-6x2+32)/(x-4)Subtract down to get a new polynomial
x2 x-4 |x3-6x2+0x+32 x3-4x2 -2x2+0x+32(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(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(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(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 0Example with a remainder x2-2x -8 x-4 |x3-6x2 x3-4x2 -2x2 -2x2+8x -8x -8x+32 -32Example with a remainder x2-2x -8 x-4 |x3-6x2 x3-4x2 -2x2 -2x2+8x -8x -8x+32 -32x-4 Is NOT a factor of x3-6x2Example with a remainder x2-2x -8 x-4 |x3-6x2 x3-4x2 -2x2 -2x2+8x -8x -8x+32 -32x-4 Is NOT a factor of x3-6x2
Synthetic DivisionSynthetic DivisionDoes exactly the same thing as polynomial long divisionFasterTakes up less spaceEasier (for me, at least)Factoring PolynomialsLets 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 likex3-6x2+32=(x-4)(.)Polynomial Divisionx3-6x2+32=(x-4)(.)In order to find the (.), I have to divide both sides by (x-4).(x3-6x2+32)/(x-4)=(.)
What is the quotient when the polynomial3x3 18x2 27x+ 162 is divided by x-3? 3x2+9x-54 3x2+9x+543x2-9x+54 3x2-9x-54None of the above is completelycorrectWhat is the quotient when the polynomial3x3 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 -162D) 3x2-9x-54 Fun Tricks with Synthetic DivisionIf you divide (x) and (x-c), then the remainder is the value of (c)
Example: (x)=3x3-18x2-27x+162
3 -9 -54 0 3 |3 -18 -27 162 9 -27 -162
Remainder is 0, so (3)=0
Fun Tricks with Synthetic DivisionIf you divide (x) and (x-c), then the remainder is the value of (c)
Example: (x)=3x3-18x2-27x+162
3 -9 -54 0 3 |3 -18 -27 162 9 -27 -162
Remainder is 0, so (3)=0
3 is a root(x-3) is a factorFun Tricks with Synthetic DivisionIf you divide (x) and (x-c), then the remainder is the value of (c)
Example: (x)=3x3-18x2-27x+162
3 -15 -42 120 1 |3 -18 -27 162 3 -15 -42
Remainder is 120, so (1)=120
Fun Tricks with Synthetic DivisionIf you divide (x) and (x-c), then the remainder is the value of (c)
Example: (x)=3x3-18x2-27x+162
3 -15 -42 120 1 |3 -18 -27 162 3 -15 -42
Remainder is 120, so (1)=120
1 is NOT a root(x-1) is NOT a factorFinal ThoughtYour book (and possibly your recitation instructor) write synthetic division upside down. Its 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 | 0Is thesame asWhich 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