Problem 52 - Intermediate Algebra

Question

Perform each division. Divide \(x^{4}+2+4 x^{2}+3 x+2 x^{3}\) by \(x^{2}+2+x\)

Step-by-Step Solution

Answer

The quotient is \(x^2 + x + 1\), and the remainder is 0.

Step 1: Arrange the Terms

First, ensure that both the dividend and the divisor polynomials are arranged in descending powers of x. The dividend is initially given as \(x^4 + 2 + 4x^2 + 3x + 2x^3\). Let's rearrange it to \(x^4 + 2x^3 + 4x^2 + 3x + 2\). The divisor, \(x^2 + 2 + x\), should be rearranged to \(x^2 + x + 2\).

Step 2: Perform Polynomial Division

Set up the division in long division format, where \(x^4 + 2x^3 + 4x^2 + 3x + 2\) is divided by \(x^2 + x + 2\). Begin the division process by dividing the leading term of the dividend, \(x^4\), by the leading term of the divisor, \(x^2\). This gives you \(x^2\).

Step 3: Multiply and Subtract

Multiply \(x^2\) by the entire divisor \(x^2 + x + 2\) to get \(x^4 + x^3 + 2x^2\). Subtract \(x^4 + x^3 + 2x^2\) from the dividend, \(x^4 + 2x^3 + 4x^2 + 3x + 2\), resulting in \(x^3 + 2x^2 + 3x + 2\).

Step 4: Repeat the Process

Divide the new leading term \(x^3\) by \(x^2\) to get \(x\). Multiply \(x\) by \(x^2 + x + 2\) to get \(x^3 + x^2 + 2x\). Subtract this from \(x^3 + 2x^2 + 3x + 2\) to obtain \(x^2 + x + 2\).

Step 5: Final Division

Now, divide \(x^2\) by \(x^2\) to get \(1\). Multiply \(1\) by \(x^2 + x + 2\) to get \(x^2 + x + 2\). Subtract this from \(x^2 + x + 2\), leaving a remainder of 0. Therefore, the division is complete.

Step 6: Conclusion of the Division

The quotient obtained from the division is \(x^2 + x + 1\) with a remainder of 0. This means that \(x^4 + 2x^3 + 4x^2 + 3x + 2\) is exactly divisible by \(x^2 + x + 2\).

Key Concepts

Long Division of Polynomials

Polynomial long division is a method just like the long division of numbers. The goal is to divide one polynomial (the dividend) by another (the divisor) to find the quotient and the remainder. First, we organize the terms of both polynomials in descending order of their degrees. This organization ensures clarity when performing each step of division.

When ready, you work from left to right, dividing the highest degree term of the dividend by the highest degree term of the divisor. This gives the first term of the quotient. Multiply the entire divisor by this term and subtract the result from the dividend.

• Think of it like peeling off layers of the polynomial, one term at a time.
• Each step simplifies the original polynomial and helps uncover the full quotient.

Continue this process, handling one term at a time, until the degrees of the remaining part of the dividend are less than the degree of the divisor. This remaining part of the polynomial is known as the remainder.

Quotient and Remainder

In polynomial division, just like with numbers, the result can be broken down into a quotient and a remainder. The quotient is what you get after the division when all possible matches are subtracted, while the remainder is what's left over if the division isn't exact.

For our example, the dividend is fully divisible by the divisor, resulting in a 0 remainder. This indicates a perfect division, showing:

• The entire dividend has been split evenly by the divisor with no leftovers.
• The final quotient is straightforwardly derived from the set of terms obtained in each step of the division.

When the remainder is nonzero, it often means that the dividend wasn't exactly divisible by the divisor using whole terms. It's an essential check at the end of every division process to ensure accuracy.

Rearranging Polynomials

Rearranging polynomials is a fundamental but often overlooked step in solving polynomial equations or performing operations like division. Each polynomial expression should be organized from the highest to the lowest degree of terms. This arrangement simplifies identifying terms you need to operate on step-by-step.

In our case, rearranging the dividend from an unordered style, such as continually presented, ensures a smooth division process.

• Start with the term with the highest degree. This should be at the beginning of your polynomial.
• The terms should progress with decreasing exponent values until reaching the constant term, if any.

This structured format helps clarify which parts of the polynomial participate in the division and guarantees you don't miss or misplace any terms in your operations.