Problem 75 - Intermediate Algebra

Question

Perform each division. \(\frac{x^{5}+3 x+2}{x^{3}+1+2 x}\)

Step-by-Step Solution

Answer

The quotient is \(x^2 - 2\) with remainder \(-x^2 + 7x + 4\).

Step 1: Ensure Polynomial Order

Before we perform polynomial division, make sure both the numerator and divisor (denominator) are written in standard form. The numerator is already standard as \(x^5 + 3x + 2\) and the divisor is \(x^3 + 2x + 1\). Check if terms in both polynomials are ordered by the descending powers of \(x\).

Step 2: Set Up Division

We are performing polynomial long division. Set up the division in a similar fashion to long division with numbers. Place \(x^5 + 3x + 2\) as the dividend inside the division symbol, and \(x^3 + 2x + 1\) as the divisor outside.

Step 3: Divide the Leading Terms

Divide the leading term of the dividend \(x^5\) by the leading term of the divisor \(x^3\). This gives us \(x^2\). Write \(x^2\) on top of the division bracket.

Step 4: Multiply and Subtract

Multiply the entire divisor \(x^3 + 2x + 1\) by \(x^2\) to get \(x^5 + 2x^3 + x^2\). Subtract this result from the dividend \(x^5 + 3x + 2\) to get \(-2x^3 - x^2 + 3x + 2\).

Step 5: Repeat the Division Process

Repeat the steps of dividing the leading term of the new polynomial \(-2x^3\) by the leading term of the divisor \(x^3\). This results in \(-2\). Write \(-2\) above the division bracket next to \(x^2\).

Step 6: Multiply and Subtract Again

Multiply the divisor \(x^3 + 2x + 1\) by \(-2\) to get \(-2x^3 - 4x - 2\). Subtract this from \(-2x^3 - x^2 + 3x + 2\) to get \(-x^2 + 7x + 4\).

Step 7: Determine Remainder and Quotient

At this point, \(-x^2 + 7x + 4\) is of lower degree than the divisor \(x^3 + 2x + 1\). We cannot divide any further and this polynomial \(-x^2 + 7x + 4\) is the remainder. The division yields \(x^2 - 2\) with a remainder of \(-x^2 + 7x + 4\).

Key Concepts

Standard Form

In algebra, writing polynomials in standard form is crucial when performing operations such as addition, subtraction, multiplication, and especially division. A polynomial is in standard form when its terms are arranged from the highest power to the lowest power of the variable. For example, the polynomial \(x^5 + 3x + 2\) is in standard form because the terms are ordered by decreasing powers of \(x\) (\(x^5\), \(x\), then the constant term \(2\)).

• Check the degree of each term, which is the exponent of \(x\).
• Arrange the terms in decreasing order based on their degree.

This organization is not just for neatness; it ensures consistent results when substituting the polynomials into various algebraic processes, especially division.

Polynomial Long Division

Polynomial long division is a systematic method for dividing one polynomial by another, similar to numerical long division. It involves dividing, multiplying, and subtracting steps in succession until the degree of the remaining polynomial (remainder) is lower than the divisor.

• First, set up the division bracket, placing the dividend inside and the divisor outside.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the entire divisor by this term and subtract the result from the dividend.
• Repeat this process with the new polynomial obtained after subtraction until reaching a remainder.

This method transforms complex polynomial division into a manageable, step-by-step sequence closely resembling the familiar process of long division in arithmetic.

Remainder

When performing polynomial division, the remainder is the polynomial that is left over once the division process cannot proceed because the degree of the remaining polynomial is less than that of the divisor. In our original example, after dividing the polynomial \(x^5 + 3x + 2\) by \(x^3 + 2x + 1\), the remainder is \(-x^2 + 7x + 4\).

• The remainder is always less in degree than the divisor.
• If the remainder is zero, the division is exact, meaning the quotient without the remainder divides the dividend perfectly.
• Otherwise, the remainder is included in the final quotient as \(\text{quotient} + \frac{\text{remainder}}{\text{divisor}}\).

Understanding the remainder concept is crucial in polynomial division because it shows how much of the dividend is not fully divisible by the divisor and affects the final expression of the division.

Quotient in Algebra

In algebra, the quotient is the result obtained when one polynomial is divided by another. Using polynomial long division, the quotient in our example comes out to be \(x^2 - 2\). This quotient represents how many times the divisor can be factored out from the original polynomial.

• Found through repeated division of leading terms, multiplication, and subtraction.
• Represents what is obtained from the division process before considering the remainder.
• Together with the remainder, it completely represents the division process such that \( \text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder} \).

The significance of the quotient lies in its ability to simplify complex polynomial expressions and solve polynomial equations by isolation of terms and roots.