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)**

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