Problem 29 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(x^{3}+64\right) \div(x+1) $$

Step-by-Step Solution

Answer

Quotient: \( x^2 - x + 1 \), Remainder: \( 63 \).

Step 1: Identify Polynomials

We are given the division \( (x^3 + 64) \div (x+1) \). Here, \( x^3 + 64 \) is the dividend and \( x+1 \) is the divisor. The task is to find the quotient by performing polynomial division.

Step 2: Set Up Long Division

Write the dividend \( x^3 + 64 \) inside the long division symbol and \( x+1 \) outside. Remember to write any missing degree term in the dividend with a coefficient of 0. So, rewrite \( x^3 + 0x^2 + 0x + 64 \).

Step 3: Divide the Leading Terms

Divide the leading term of the dividend \( x^3 \) by the leading term of the divisor \( x \). \( \frac{x^3}{x} = x^2 \). Place \( x^2 \) on top of the division bar.

Step 4: Multiply and Subtract

Multiply the entire divisor \( x+1 \) by \( x^2 \), giving \( x^3 + x^2 \). Subtract this from the current dividend \( x^3 + 0x^2 + 0x + 64 \). The result is \( -x^2 + 0x + 64 \).

Step 5: Repeat the Process

Divide the new leading term \( -x^2 \) by \( x \), giving \( -x \). Place \( -x \) above the division bar next to \( x^2 \). Multiply \( -x \) by \( x+1 \) to get \( -x^2 - x \). Subtract to obtain \( x + 64 \).

Step 6: Final Division Step

Divide the new leading term \( x \) by \( x \), resulting in \( 1 \). Place \( 1 \) on top. Multiply \( 1 \) by \( x+1 \) to get \( x + 1 \), and subtract it from \( x + 64 \) to leave a remainder of \( 63 \).

Step 7: Write the Quotient and Remainder

The quotient is \( x^2 - x + 1 \) and the remainder is \( 63 \). Thus, the division can be expressed as \( x^3 + 64 = (x+1)(x^2 - x + 1) + 63 \).

Key Concepts

Long Division of Polynomials

Long division of polynomials is a method similar to the long division used with numbers. It involves dividing a polynomial, known as the dividend, by another polynomial, known as the divisor. This technique is efficient for dividing any polynomials and can be used to find both a quotient and a remainder.

When performing long division with polynomials, it is important to set everything up carefully before you begin.

• Rewrite the dividend including all missing powers with coefficients of 0. This keeps everything in order as you work through the division process.
• The divisor goes outside of the division symbol as you would in ordinary long division with numbers.

Each step involves focusing on the leading terms and working through the polynomial step by step, subtracting products just as you would when dividing numbers. It may seem a bit complex at first, but with practice, long division of polynomials becomes a straightforward and logical process.

Dividing Polynomials

Dividing polynomials, like long division, requires you to carefully divide the polynomials term by term. The process follows a simple set of steps that mirrors the numerical long division method.

Start by taking the leading term of the dividend and dividing it by the leading term of the divisor. This will give you the first term of the quotient.

• Multiply the entire divisor by this term and subtract the resulting product from the dividend.
• Move on to the next term of the result and repeat the process until all terms of the dividend have been examined.

This iterative approach ensures that each term in the dividend has been accounted for, and you systematically remove those contributions in the course of finding the quotient.

Polynomial Remainder

In polynomial division, just like number division, you often end up with a remainder. The remainder is what is left after dividing the polynomial as much as possible.

To identify the remainder, you continue dividing the dividend term by term, as described in the previous step, until you can no longer divide because the degree of the remaining polynomial is less than that of the divisor. At that point, whatever is left is the remainder.

• The remainder can be zero, which means that the divisor perfectly divides the dividend.
• If a remainder is present, it is always of a lower degree than the divisor, ensuring that division stops when no further division of significant degree is possible.

Understanding the role of the remainder helps in grasping how polynomials can be fully expressed in the terms of a divisor and quotient, which enhances the clarity when modeling or solving equations involving polynomials.