Dividing Polynomials: (8x^3  3x + 1) ÷ (4x^3 + x^2  2x  3)
This article will guide you through the process of dividing the polynomial (8x^3  3x + 1) by (4x^3 + x^2  2x  3) using polynomial long division.
Understanding Polynomial Long Division
Polynomial long division is similar to the long division of numbers. We aim to find a quotient polynomial that, when multiplied by the divisor polynomial, results in the dividend polynomial.
The Steps

Set up the division:
 Write the dividend (8x^3  3x + 1) inside the division symbol.
 Write the divisor (4x^3 + x^2  2x  3) outside the division symbol.
 Remember to include placeholders for missing terms in both the dividend and divisor with a coefficient of 0.

Divide the leading terms:
 Focus on the leading terms of the dividend and divisor (8x^3 and 4x^3).
 Divide the leading term of the dividend by the leading term of the divisor: (8x^3) ÷ (4x^3) = 2.
 Write this result (2) as the first term of the quotient above the division symbol.

Multiply the quotient term by the divisor:
 Multiply the quotient term (2) by the entire divisor (4x^3 + x^2  2x  3) to get: 2(4x^3 + x^2  2x  3) = 8x^3 + 2x^2  4x  6

Subtract the result from the dividend:
 Write the result of the multiplication (8x^3 + 2x^2  4x  6) below the dividend, aligning terms with the same powers of x.
 Subtract the two polynomials:
(8x^3  3x + 1)  (8x^3 + 2x^2  4x  6)  2x^2 + x + 7

Bring down the next term:
 Bring down the next term from the dividend (3x) to create the new dividend: 2x^2 + x + 7  3x = 2x^2  2x + 7

Repeat steps 25:
 Repeat the process, focusing on the new dividend (2x^2  2x + 7) and the divisor (4x^3 + x^2  2x  3).
 Since the leading term of the new dividend (2x^2) has a lower degree than the leading term of the divisor (4x^3), the quotient term is 0.
 Therefore, we write 0 in the quotient above the division symbol, and we continue with the new dividend: 2x^2  2x + 7.

Continue until the degree of the remainder is less than the degree of the divisor:
 We continue the process until the degree of the remainder is less than the degree of the divisor.
 In this case, the degree of the remainder (2) is less than the degree of the divisor (3).
Solution
The final result of the polynomial division is:
(8x^3  3x + 1) ÷ (4x^3 + x^2  2x  3) = 2 + (2x^2  2x + 7) / (4x^3 + x^2  2x  3)
This means that the quotient is 2, and the remainder is 2x^2  2x + 7.
Conclusion
Polynomial long division can be a complex process, but by breaking it down into smaller steps and remembering to align terms with the same powers of x, you can successfully divide polynomials. Remember to always check your work and ensure that the degree of the remainder is less than the degree of the divisor.