Problem 52 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(4 x^{3}-2 x^{2}+7 x-5\right) \div\left(x^{2}+2\right) $$

Step-by-Step Solution

Answer

Quotient: \(4x - 2\), Remainder: \(x - 1\).

Step 1: Set Up the Division

We will perform polynomial long division. The dividend is \(4x^3 - 2x^2 + 7x - 5\) and the divisor is \(x^2 + 2\). Write them in long division form.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(4x^3\) by the leading term of the divisor \(x^2\). This gives us \(4x\). Write \(4x\) as the first term of the quotient.

Step 3: Multiply and Subtract

Multiply the entire divisor \(x^2 + 2\) by \(4x\) to get \(4x^3 + 8x\). Subtract this from the original dividend \(4x^3 - 2x^2 + 7x - 5\) to get \( -2x^2 - x - 5\).

Step 4: Repeat the Process

Now divide the leading term \(-2x^2\) of the new dividend by the leading term \(x^2\) of the divisor to get \(-2\). Add \(-2\) to the quotient.

Step 5: Multiply and Subtract Again

Multiply \(-2\) by \(x^2 + 2\) to get \(-2x^2 - 4\). Subtract this from the current dividend \(-2x^2 - x - 5\), giving us a remainder of \(x - 1\).

Step 6: Form the Final Result

The quotient is \(4x - 2\) and the remainder is \(x - 1\). Thus, the division can be expressed as \(4x-2 + \frac{x - 1}{x^2 + 2}\).

Key Concepts

Understanding the Dividend

The dividend is the polynomial that you need to divide by another polynomial. In this exercise, the dividend is given as \(4x^3 - 2x^2 + 7x - 5\). It contains different terms, ranging from third-degree down to zero degrees. Each term in the dividend represents a different power of the variable \(x\), and the specific aim is to break down these terms through the division process.

When performing polynomial long division, set up your dividend inside the division symbol similar to long division with numbers. This provides a structural framework that helps in systematically dividing every term of the dividend by the divisor. This ensures that you fully simplify the polynomial.

Remember: In polynomial division, keep the terms in descending power order, and make sure no powers are missing between the highest and lowest, even if it means adding terms with zero coefficients.

Defining the Divisor

The divisor in polynomial division is the polynomial you are dividing by. In our case, the divisor is \(x^2 + 2\). To successfully perform polynomial long division, understanding your divisor is crucial, as it determines the steps throughout the division process.

When setting up the problem, write the divisor outside the long division symbol. Each time you subtract during the division, this entire polynomial—or a multiple of it—will be involved.

• The leading term of the divisor primarily guides you in dividing the terms from the dividend.
• Always ensure the leading term is not zero to avoid computational errors.

This understanding allows for an accurate determination of how many times the divisor fits into each term of the dividend.

Discovering the Quotient

The quotient in polynomial division tells you how many times the divisor can be completely fitted into the various terms of the dividend. After you complete each division cycle, a new term is added to the quotient.

In this example, the final quotient obtained is \(4x - 2\). To reach this result, the process involves systematically dividing each leading term of the updated dividend, step by step:

• For the first cycle, \(4x^3\) was divided by \(x^2\) resulting in \(4x\).
• For the second cycle, \(-2x^2\) was divided by \(x^2\) resulting in \(-2\).

Write these results cumulatively to form the complete quotient above the division line.
If there are remaining terms after all cycles, they form part of the remainder.

Understanding the Remainder

Once the division of all terms of the dividend is complete, any leftover terms form the remainder. In our example, the remainder is \(x - 1\).

The remainder represents what is left of the dividend that cannot fully accommodate another complete divisor cycle. If the remainder is zero, it means the divisor fits perfectly into the dividend. If not, the remainder must be expressed as a fraction over the original divisor.

Our exercise results in the quotient \(4x - 2\) and a remainder of \(x - 1\), leading to the expression:\[ 4x - 2 + \frac{x - 1}{x^2 + 2} \]Recalling that the degree of the remainder should always be less than the degree of the divisor ensures the division process is complete.