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asked Dec 17, 2014 in Calculus Answers by anndreatta (160 points)

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2 Answers

I think the question should read: 3x^5-15x^4+4x^3+11x^2-9x+2 divided by x^2-5x+2. The polynomial can be rewritten: 3x^5-15x^4+6x^3-2x^3+10x^2x-4x+x^2-5x+2= 3x^3(x^2-5x+2)-2x(x^2-5x+2)+(x^2-5x+2)=(3x^3-2x+1)(x^2-5x+2). So when we divide by x^2-5x+2, we're left with 3x^3-2x+1. You get the same result using algebraic long division. The clue I found in the first two terms, which happened to be 3x^2 times the first two terms of the divisor; also I suspected that, since the polynomial and the divisor both ended in +2, the last number in the quotient would also be 1. All that was left was to find the middle term of the quotient.
answered Dec 23, 2014 by Rod Top Rated User (582,800 points)

Polynomial Division: (3x^5 - 15x^4 + 4x^3 + 11x^2 - 9x + 2)/ (x^2 - 5x + 2)

Write problem in special format:

x^2 - 5x + 2/ 3x^5 - 15x^4 + 4x^3 + 11x^2 - 9x + 2

Step 1: Divide the leading term of the dividend by the leading term of the divisor: 3x^5/x^2 = 3x^3

Write calculated result in upper part table: Multiply it by the divisor:

3x^3 (x^2 - 5x + 2) = 3x^5 - 15x^4 + 6x^3

Subtract dividend from obtain result:

(3x^5 - 15x^4 + 4x^3 + 11x^2 - 9x + 2) - (3x^5 - 15x^4 + 6x^3)

= - 2x^3 + 11x^2 - 9x + 2

 

x^2 - 5x + 2/3x^5 - 15x^4 +4x^3 +11x^2 -9x +2 = 3x^5/x^2 = 3x^3

- 3x^5 -15x^4 +6x^3 3x^3 (x^2 - 5x + 2) = 3x^5 - 15x^4 + 6x^3

- 2x^3 +11x^2 -9x +2

Step 2: Divide the leading term of the obtained remainder by the leading term of the divisor: - 2x^3/x^2 = -2x

Write down the calculated result in the upper part of the table: Multiply it by the divisor:

- 2x ( x^2 - 5x + 2) = -2x^3 + 10x^2 - 4x

Subtract the remainder from obtaining results:

( - 2x^3 + 11x^2 - 9x + 2) - ( - 2x^3 + 10x^2 - 4x) = x^2 - 5x + 2

 

x^2 - 5x + 2/ 3x^5 -15x^4 + 4x^3 +11x^2 -9x +2 = 3x^3 -2x

- 3x^5 -15x^4 +6x^3

-2x3 +11x^2 -9x +2 -2x^3/x^2 = -2x

-2x3 +10x^2 -4x - 2x (x^2 - 5x + 2) = -2x^3 + 10x^2 - 4x

x^2 -5x +2

 

Step 3: Divide the leading term of the obtained remainder by the leading term of the divisor: x^2/x^2 = 1

Write down the calculated results in the upper part of table. Multiply by the divisor: 1 (x^2 - 5x + 2) = x^2 - 5x + 2

Subtract the remainder from obtained result: (x^2 - 5x + 2) - (x^2 - 5x + 2) =

x^2 - 5x + 2/3x^5 -15x^4 +4x^3 +11x^2 -9x +2

- 3x^5 -15x^4 +6x^3

- 2x^3 +11x^2 -9x +2

- 2x^3 +10x^2 -4x

x^2 -5x +2 x^2/x^2 = 1

- x^2 -5x +2

0 1 (x^2 - 5x + 2) = x^2 - 5x + 2

 

Since the degree of the remainder is less than the degree of the divisor, then were done:

Therefore, 3x^5 - 15x^4 + 4x^3 + 11x^2 - 9x + 2/x^2 - 5x + 2

= 3x^3 - 2x + 1

+ 0

x^2 - 5x + 2 = 3x^3 - 2x + 1

 

 

Answer:

3x^5 - 15x^4 + 4x^3 + 11x^2 - 9x + 2/x^2 - 5x + 2

= 3x^3 - 2x + 1

+ 0

x^2 - 5x + 2

Answer:  (3x^3 - 2x + 1)

 

 

answered Jul 29 by anonymous

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