Problem 21 - Algebra and Trigonometry

Question

\(15-24=\) Find the quotient and remainder using long division. $$ \frac{6 x^{3}+2 x^{2}+22 x}{2 x^{2}+5} $$

Step-by-Step Solution

Answer

Quotient: \(3x + 1\), Remainder: \(7x - 5\).

Step 1: Set up the Division

We are asked to divide the polynomial \(6x^3 + 2x^2 + 22x\) by \(2x^2 + 5\) using long division. We'll set it up just like a numerical long division, placing \(6x^3 + 2x^2 + 22x\) inside the division bracket and \(2x^2 + 5\) outside.

Step 2: Divide the Leading Terms

Look at the leading term of the dividend \(6x^3\) and the leading term of the divisor \(2x^2\). Divide these: \(\frac{6x^3}{2x^2} = 3x\). This will be the first term of our quotient.

Step 3: Multiply and Subtract

Now multiply the entire divisor \(2x^2 + 5\) by the first term of our quotient \(3x\), which gives \((3x)(2x^2 + 5) = 6x^3 + 15x\). Subtract this from \(6x^3 + 2x^2 + 22x\), resulting in \((6x^3 + 2x^2 + 22x) - (6x^3 + 15x) = 2x^2 + 7x\).

Step 4: Repeat the Process

Next, divide the leading term of the new polynomial \(2x^2\) by the leading term of the divisor \(2x^2\), which gives \(1\). This is the next term in our quotient. Multiply the entire divisor by \(1\), giving \(2x^2 + 5\). Subtract this from \(2x^2 + 7x\), resulting in \((2x^2 + 7x) - (2x^2 + 5) = 7x - 5\).

Step 5: Final Division and Remainder

Try to divide the leading term \(7x\) by \(2x^2\). Since the degree of \(7x\) is less than that of \(2x^2\), no further division is possible. Thus, \(7x - 5\) is the remainder.

Step 6: Write the Solution

The division gives us a quotient of \(3x + 1\) and a remainder of \(7x - 5\).

Key Concepts

Long Division

Polynomial long division is a method for dividing a polynomial by another polynomial, much like the long division of numbers that you may already be familiar with. In polynomial long division, we align terms based on descending powers of the variable.

Here's a quick rundown:

• Set up the division with the larger polynomial (the dividend) inside the division symbol and the smaller polynomial (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 first term of the quotient and subtract the result from the original dividend.
• Repeat the process with the new polynomial until the remainder becomes less than the degree of the divisor.

The goal is to systematically reduce the problem until you get a remainder that cannot be divided further by the divisor. This ensures you have found both the quotient and the remainder of the division.

Quotient

The quotient in polynomial division is the resulting polynomial that you get when you divide the dividend by the divisor. In our example, after performing the long division on the given polynomials, we arrived at the quotient of the division: \(3x + 1\).

Here’s how the quotient is determined:

• Begin by dividing the leading term of the dividend by the leading term of the divisor. For instance, \(\frac{6x^3}{2x^2} = 3x\), which is the first term of our quotient.
• Multiply the entire divisor by this first quotient term and subtract it from the original polynomial.
• Repeat these steps with any remaining terms after subtraction.

The quotient represents how many times the divisor fits into the dividend perfectly without exceeding it. Hence, after every subtraction, if terms still exist, the same process is repeated to find the next term of the quotient.

Remainder

In polynomial division, the remainder is what is left over after performing the long division process when no more terms of the quotient can be obtained. In our case, after dividing, the remainder turned out to be \(7x - 5\).

Understanding the remainder involves:

• Attempting to divide the remaining polynomial by the divisor.
• If the degree of the remaining polynomial is less than that of the divisor, then no further division is possible—this leftover portion is the remainder.

The remainder, similar to numeric division, is what remains when the divisor no longer "fits" into the dividend. It serves as an essential component of the division, represented together with the quotient to explain the entire division process: Dividend = Divisor × Quotient + Remainder.

Leading Term

The leading term is the term in a polynomial with the highest degree, which usually means it is the term with the largest exponent of the variable. Identifying the leading term is crucial in the division process because it determines how the division proceeds.

During polynomial long division:

• We use the leading term of both the dividend and the divisor to calculate the first term of the quotient.
• This ensures that we are systematically eliminating the terms in the dividend from highest to lowest power.
• In our example, at first, we took the leading term \(6x^3\) from the dividend and divided by the leading term \(2x^2\) from the divisor.

By focusing on the leading terms, we streamline the process of finding each term of the quotient, progressively simplifying the polynomial until it’s completely divided or until only the remainder is left.