Problem 45 - Algebra for College Students

Question

Perform the indicated divisions. $$ \left(2 n^{4}+3 n^{3}-2 n^{2}+3 n-4\right) \div\left(n^{2}+1\right) $$

Step-by-Step Solution

Answer

The quotient is \( 2n^2 + 3n - 4 \).

Step 1: Set Up Long Division

The problem can be solved using polynomial long division. We will divide \( 2n^4 + 3n^3 - 2n^2 + 3n - 4 \) by \( n^2 + 1 \). Start by writing the dividend \( 2n^4 + 3n^3 - 2n^2 + 3n - 4 \) and the divisor \( n^2 + 1 \).

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \( 2n^4 \) by the leading term of the divisor \( n^2 \) to get the first term of the quotient, \( 2n^2 \).

Step 3: Multiply and Subtract

Multiply the entire divisor \( n^2 + 1 \) by the term \( 2n^2 \) to get \( 2n^4 + 2n^2 \). Subtract this result from the original dividend to get a new polynomial: \( 3n^3 - 4n^2 + 3n - 4 \).

Step 4: Divide the New Leading Terms

Divide the new leading term \( 3n^3 \) by \( n^2 \) to get the next term of the quotient, \( 3n \).

Step 5: Multiply and Subtract Again

Multiply the divisor \( n^2 + 1 \) by \( 3n \) to obtain \( 3n^3 + 3n \). Subtract this from the current dividend to obtain a new polynomial: \( -4n^2 - 4 \).

Step 6: Divide Once More

Divide the new leading term \( -4n^2 \) by \( n^2 \) to get the last term of the quotient, \( -4 \).

Step 7: Final Multiply and Subtract

Multiply \( n^2 + 1 \) by \( -4 \) to get \( -4n^2 - 4 \). Subtract this from the current dividend to obtain the remainder: \( 0 \).

Step 8: Write the Final Answer

Since the remainder is 0, the division is exact. The quotient of \( \left(2 n^{4}+3 n^{3}-2 n^{2}+3 n-4\right) \div \left(n^{2}+1\right) \) is \( 2n^2 + 3n - 4 \).

Key Concepts

Quotient

In polynomial long division, the **quotient** is the result you obtain after dividing one polynomial by another. Think of it as the answer to a division problem in arithmetic. When you divide the polynomial \( 2n^4 + 3n^3 - 2n^2 + 3n - 4 \) by \( n^2 + 1 \), the quotient is \( 2n^2 + 3n - 4 \).
This means that \( 2n^2 + 3n - 4 \) is the polynomial that is multiplied by the divisor to yield the dividend, assuming no remainder is left.

• To find each term of the quotient, divide the leading term of the current dividend by the leading term of the divisor.
• Then, multiply the entire divisor by this term and subtract from the current polynomial.

Following these steps helps construct the entire quotient one term at a time, until the division process is complete.

Remainder

The **remainder** is what remains after you complete the polynomial long division. It's similar to what you have left when dividing integers if the division isn’t exact. In this exercise, after performing all the necessary steps, the remainder is zero.
This tells us that \( 2n^4 + 3n^3 - 2n^2 + 3n - 4 \) is evenly divisible by \( n^2 + 1 \), and no extra terms are needed to express the original polynomial fully. If there were a non-zero remainder, you would represent it as a fraction added to the quotient, where the remainder becomes the numerator and the divisor becomes the denominator.

• The presence of a zero remainder means the quotient exactly divides the dividend without any leftover.
• A positive remainder implicates an incomplete division that can be expressed as an additional simplified fraction.

Leading Term

The **leading term** in both the dividend and divisor plays a pivotal role in polynomial long division. When dividing, you always begin by comparing these leading terms. In this problem, the leading term of the dividend is \( 2n^4 \) and the divisor’s leading term is \( n^2 \).
The leading term indicates the highest power of the variable present in the polynomial and it essentially "leads" the polynomial.

• During division, the leading term of the dividend is divided by the leading term of the divisor to determine the first term of the quotient.
• After completing each division step, the leading term changes, and the process repeats on the new polynomial formed after subtraction.

Understanding this concept ensures that you correctly determine each subsequent term of the quotient as you move through the division.

Dividend and Divisor

In polynomial division, the **dividend** is the polynomial that you are dividing, and the **divisor** is the polynomial by which you are dividing. In this exercise, \( 2n^4 + 3n^3 - 2n^2 + 3n - 4 \) is the dividend, and \( n^2 + 1 \) is the divisor.
The setup for a polynomial long division problem is to write the dividend inside the division bracket and place the divisor outside.

• It's crucial to arrange the terms of both the dividend and divisor in descending order according to the powers of the variables.
• The first term of the dividend must have a degree equal to or higher than the first term of the divisor, otherwise division in its current form isn't possible without modifying the problem.

Proper understanding of the roles of dividend and divisor is fundamental in smoothly executing the division and avoiding errors in finding the quotient.