Problem 21 - Precalculus Mathematics for Calculus

Question

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: 2x.

Step 1: Setup the Long Division

Write down the dividend and divisor in the long division format. The dividend is the polynomial in the numerator, \(6x^3 + 2x^2 + 22x\), and the divisor is \(2x^2 + 5\).

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(6x^3\) by the leading term of the divisor \(2x^2\). This yields \(3x\). Write \(3x\) above the division bar.

Step 3: Multiply and Subtract

Multiply the entire divisor \(2x^2 + 5\) by \(3x\) to get \(6x^3 + 15x\). Subtract this result from the current dividend \(6x^3 + 2x^2 + 22x\) to get the new dividend \(2x^2 + 7x\).

Step 4: Repeat Division

Divide the new leading term of the dividend \(2x^2\) by the leading term of the divisor \(2x^2\), which yields \(1\). Write \(1\) above the division bar next to \(3x\).

Step 5: Multiply and Subtract Again

Multiply the entire divisor \(2x^2 + 5\) by \(1\) to get \(2x^2 + 5\). Subtract this from the current dividend \(2x^2 + 7x\) to get the remainder \(2x\).

Step 6: Final Quotient and Remainder

The division is complete as the remaining term degree \(2x\) is less than the degree of the divisor. Therefore, the quotient is \(3x + 1\) and the remainder is \(2x\).

Key Concepts

Quotient and Remainder

In mathematics, the quotient and remainder are the results we get from dividing two numbers or expressions. When dealing with polynomials, the concept is similar to regular division, but instead of numbers, we work with polynomials.

• The quotient is the result obtained by dividing the dividend by the divisor until no further divisions are possible without going into decimals or negatives.
• The remainder is what is left over when no further division is possible. It has a degree lower than the divisor.

Understanding the quotient and remainder in polynomial division is crucial because it helps us express rational expressions more simply. When a polynomial is divided by another polynomial, the remainder must always be of a lower degree than the divisor. If not, the division process must continue. In our exercise, we obtained a quotient of \(3x + 1\) and a remainder of \(2x\). The remainder, \(2x\), is of lower degree than the divisor, making our solution complete. This emphasizes the importance of achieving a remainder that cannot be further divided by the divisor.

Polynomial Division

Polynomial division is a method used to divide one polynomial by another, similar to the long division process used with numbers. It is particularly useful for simplifying complex rational expressions. The process involves several steps, which include dividing, multiplying, and subtracting until the remainder is of a smaller degree than the divisor.To carry out polynomial division, follow these steps:

• Set up the long division by writing the dividend and divisor.
• Divide the leading term of the dividend by the leading term of the divisor to find the first term of the quotient.
• Multiply the whole divisor by this quotient term and subtract the result from the dividend.
• Repeat the process with the new resulting polynomial until the remainder has a lower degree than the divisor.

In the exercise provided, the polynomial division method was applied to divide \(6x^3 + 2x^2 + 22x\) by \(2x^2 + 5\). By successfully completing these steps, we find that the quotient is \(3x + 1\) and the remainder is \(2x\). Mastery of this division process is key to understanding higher-level algebra issues and is an essential tool in various fields of mathematics.

Rational Expressions

Rational expressions are fractions where both the numerator and the denominator are polynomials. They are similar to rational numbers, which are of the form \(\frac{a}{b}\), where \(a\) and \(b\) are integers. However, in rational expressions, polynomials replace the integers. Simplifying these expressions often involves operations like polynomial long division.Rational expressions can be complicated and messier than rational numbers due to the presence of variables. Here is why polynomial division is helpful:

• Simplifies complex expressions by dividing polynomials to reduce them to simpler forms.
• Helps in breaking down parts of functions which can aid in integration and differentiation in calculus.
• Aids in solving equations by reducing fractions to their simplest form.

From our example, using polynomial division allowed us to rewrite the rational expression \(\frac{6x^3 + 2x^2 + 22x}{2x^2 + 5}\) into a simpler form with quotient \(3x + 1\) and remainder \(2x\). This process makes it easier to analyze and solve further problems. It forms the basis for understanding complex relationships between variables in advanced mathematical concepts.