Long Division of Polynomials: (x^5 + 7) / (x^3  1)
Long division of polynomials is a method used to divide one polynomial by another. This process can be used to simplify expressions, solve equations, and find factors of polynomials.
Let's work through the division of (x^5 + 7) / (x^3  1) stepbystep.
Setting up the Division

Organize the polynomials: Arrange the terms of both polynomials in descending order of their exponents. If any terms are missing, fill in the gaps with a coefficient of 0.
x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7 __________________________________ x^3  1 

Start the division: Divide the leading term of the dividend (x^5) by the leading term of the divisor (x^3). This gives us x^2. Write this term above the line, aligned with the x^5 term.
x^2 __________________________________ x^3  1  x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7

Multiply the divisor by the term just written: Multiply (x^3  1) by x^2. This gives us x^5  x^2. Write this result below the dividend, aligning the terms.
x^2 __________________________________ x^3  1  x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7 x^5  x^2

Subtract: Subtract the terms you just wrote from the dividend. Remember to change the signs of the terms you are subtracting.
x^2 __________________________________ x^3  1  x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7 x^5  x^2  0x^4 + 0x^3 + x^2 + 0x + 7
Continue the Process

Bring down the next term: Bring down the next term of the dividend (0x) to the result.
x^2 __________________________________ x^3  1  x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7 x^5  x^2  0x^4 + 0x^3 + x^2 + 0x + 7

Repeat the steps: Divide the leading term of the new dividend (0x^4) by the leading term of the divisor (x^3). This gives us 0. Write this term above the line, aligned with the x^4 term.
x^2 + 0 __________________________________ x^3  1  x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7 x^5  x^2  0x^4 + 0x^3 + x^2 + 0x + 7

Multiply and subtract: Multiply (x^3  1) by 0. Write this result below the new dividend, align the terms and subtract.
x^2 + 0 __________________________________ x^3  1  x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7 x^5  x^2  0x^4 + 0x^3 + x^2 + 0x + 7 0x^4 + 0x^3 + 0x  x^2 + 0x + 7

Continue: Repeat the process until the degree of the remaining dividend is less than the degree of the divisor.
x^2 + 0 + 1 __________________________________ x^3  1  x^5 + 0x^4 + 0x^3 + 0x^2 + 0x + 7 x^5  x^2  0x^4 + 0x^3 + x^2 + 0x + 7 0x^4 + 0x^3 + 0x  x^2 + 0x + 7 x^2  1  0x + 8
Final Result
The final result of the long division is:
(x^5 + 7) / (x^3  1) = x^2 + 1 + (8 / (x^3  1))
The term (8 / (x^3  1)) is the remainder of the division. This means that the original polynomial can be expressed as the sum of the quotient (x^2 + 1) and the remainder (8 / (x^3  1)) multiplied by the divisor (x^3  1).