(3x^4+15x^3+3x^2-12x+4)/(x^2+4x-2)

Jasonbradley

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Jun 16, 2024

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Dividing Polynomials: (3x^4+15x^3+3x^2-12x+4)/(x^2+4x-2)

This article will walk you through the process of dividing the polynomial 3x^4+15x^3+3x^2-12x+4 by x^2+4x-2. We'll use polynomial long division to achieve this.

Understanding Polynomial Long Division

Polynomial long division follows a similar process to the long division you learned in elementary school, but with polynomials instead of numbers. The goal is to find a quotient and remainder that satisfy the equation:

Dividend = Divisor * Quotient + Remainder

In our case:

• Dividend: 3x^4+15x^3+3x^2-12x+4
• Divisor: x^2+4x-2

Performing the Division

1. Set up the division: Write the dividend and divisor in a long division format:

             __________
x^2+4x-2 | 3x^4+15x^3+3x^2-12x+4 

2. Focus on the leading terms: Divide the leading term of the dividend (3x^4) by the leading term of the divisor (x^2). This gives us 3x^2.

3. Multiply and subtract: Multiply the divisor (x^2+4x-2) by 3x^2 and subtract the result from the dividend:

             3x^2       
x^2+4x-2 | 3x^4+15x^3+3x^2-12x+4 
             -(3x^4+12x^3-6x^2)
             -------------------
                     3x^3+9x^2-12x 

4. Bring down the next term: Bring down the next term of the dividend (-12x).

5. Repeat steps 2-4: Divide the new leading term (3x^3) by the leading term of the divisor (x^2) to get 3x. Multiply the divisor by 3x and subtract:

             3x^2+3x      
x^2+4x-2 | 3x^4+15x^3+3x^2-12x+4 
             -(3x^4+12x^3-6x^2)
             -------------------
                     3x^3+9x^2-12x 
                     -(3x^3+12x^2-6x)
                     -------------------
                             -3x^2-6x+4

6. Repeat again: Bring down the last term (4) and repeat steps 2-4:

             3x^2+3x-3      
x^2+4x-2 | 3x^4+15x^3+3x^2-12x+4 
             -(3x^4+12x^3-6x^2)
             -------------------
                     3x^3+9x^2-12x 
                     -(3x^3+12x^2-6x)
                     -------------------
                             -3x^2-6x+4
                             -(-3x^2-12x+6)
                             -----------------
                                     6x-2

7. The remainder: We've reached a point where the degree of the remainder (6x-2) is less than the degree of the divisor (x^2+4x-2). This means we've finished the division.

Conclusion

Therefore, the quotient is 3x^2 + 3x - 3 and the remainder is 6x - 2. We can express this as:

(3x^4+15x^3+3x^2-12x+4) / (x^2+4x-2) = 3x^2 + 3x - 3 + (6x - 2) / (x^2 + 4x - 2)

This shows that the original polynomial can be expressed as the sum of the quotient and a term involving the remainder divided by the divisor.