%2F(x%5E2%2B2x%2B2).jpeg "(2x^4-x^2+3x+1)/(x^2+2x+2)")
Dividing Polynomials: A Step-by-Step Guide
This article will guide you through the process of dividing the polynomial (2x^4 - x^2 + 3x + 1) by (x^2 + 2x + 2).
Understanding Polynomial Division
Polynomial division is similar to long division of numbers. We use a systematic approach to find the quotient and remainder of dividing one polynomial by another.
Steps to Divide Polynomials
-
Set Up: Arrange both polynomials in descending order of their exponents. If any terms are missing, use a placeholder with a coefficient of zero.
x^2 + 2x + 2 | 2x^4 + 0x^3 - x^2 + 3x + 1 -
Divide the Leading Terms: Divide the leading term of the dividend (2x^4) by the leading term of the divisor (x^2). This gives us 2x^2.
x^2 + 2x + 2 | 2x^4 + 0x^3 - x^2 + 3x + 1 2x^2 -
Multiply and Subtract: Multiply the quotient term (2x^2) by the entire divisor (x^2 + 2x + 2) and subtract the result from the dividend.
x^2 + 2x + 2 | 2x^4 + 0x^3 - x^2 + 3x + 1 2x^2 -(2x^4 + 4x^3 + 4x^2) ------------------- -4x^3 - 5x^2 + 3x -
Bring Down the Next Term: Bring down the next term from the dividend (3x).
x^2 + 2x + 2 | 2x^4 + 0x^3 - x^2 + 3x + 1 2x^2 -(2x^4 + 4x^3 + 4x^2) ------------------- -4x^3 - 5x^2 + 3x -
Repeat Steps 2-4: Repeat the process of dividing the leading term, multiplying, subtracting, and bringing down terms until the degree of the remainder is less than the degree of the divisor.
x^2 + 2x + 2 | 2x^4 + 0x^3 - x^2 + 3x + 1 2x^2 - 4x + 3 -(2x^4 + 4x^3 + 4x^2) ------------------- -4x^3 - 5x^2 + 3x -(-4x^3 - 8x^2 - 8x) ------------------- 3x^2 + 11x + 1 -(3x^2 + 6x + 6) ------------------- 5x - 5 -
Express the Result: The final result is expressed as:
Quotient: 2x^2 - 4x + 3 Remainder: 5x - 5
Therefore, we can write the division as:
(2x^4 - x^2 + 3x + 1) / (x^2 + 2x + 2) = 2x^2 - 4x + 3 + (5x - 5) / (x^2 + 2x + 2)