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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.
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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

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)

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