Problem 13 - College Algebra

Question

Divide using long division. State the quotient, \(q(x),\) and the remainder, \(r(x)\). $$\frac{6 x^{3}+13 x^{2}-11 x-15}{3 x^{2}-x-3}$$

Step-by-Step Solution

Answer

The quotient \(q(x)\) is \(2x+\frac{5}{3}+\frac{1}{3}\) and the remainder \(r(x)\) is \(-9\).

Step 1: Set up the long division

Set up the long division in same way as you would for numerical long division. Write down \(6x^{3}+13x^{2}-11x-15\) in the dividend place and \(3x^{2}-x-3\) in the divisor place of the long division structure.

Step 2: Perform the first step of division

In the first step we need to divide the first term of the given dividend, \(6x^{3}\), by the first term of the divisor, \(3x^{2}\), to get the first term of the quotient. So \( \frac{6x^{3}}{3x^{2}} = 2x\). Write down \(2x\) as the first term of the quotient.

Step 3: Multiply and Subtract

Now, multiply the divisor \(3x^{2}-x-3\) by \(2x\) that we just obtained, and subtract the result from the original polynomial \(6x^{3}+13x^{2}-11x-15\). After subtracting, we get a new polynomial \(5x^{2}-3x-15\). This is our new dividend.

Step 4: Perform the next step of division

Repeat the steps with this newly obtained dividend. Divide the first term of the new dividend, \(5x^{2}\), by the first term of the divisor, \(3x^{2}\). So, \( \frac{5x^{2}}{3x^{2}} = \frac{5}{3}\). Write down \(+\frac{5}{3}\) next to \(2x\) in the quotient.

Step 5: Multiply and Subtract

Multiply the divisor \(3x^{2}-x-3\) by the newly found term of the quotient, \(\frac{5}{3}\), and subtract the result from the current dividend \(5x^{2}-3x-15\). After subtracting, we get \(x-10\). This is our new dividend.

Step 6: Perform the next step of division

Again, divide the first term of the dividend, \(x\), by the first term of the divisor, \(3x^{2}\) to get \(+\frac{1}{3}\). Write down \(+\frac{1}{3}\) next to \(\frac{5}{3}\) in the quotient.

Step 7: Multiply and get the Remainder

Now multiply the entire divisor, \(3x^{2}-x-3\), by \(\frac{1}{3}\) and subtract the result from the latest dividend, \(x-10\). After subtracting, we get the final polynomial which represents the remainder. The remainder in this case is \(-10+1 = -9\).

Step 8: Write down the final quotient and the remainder

Finally, the division process is finished, and we have found both the quotient and the remainder. The quotient \(q(x) = 2x+\frac{5}{3}+\frac{1}{3}\) and the remainder \(r(x) = -9\).

Key Concepts

Quotient and Remainder

When dividing polynomials, just like dividing numbers, we end up with a quotient and a remainder. The quotient is the result of the division, which is the expression you obtain when the divisor fully 'fits into' the dividend. On the other hand, the remainder is what’s left over when you've done all the dividing you can. In the context of polynomial long division, these terms are essential. The goal is to express the dividend as a combination of the divisor and the remainder, such that:\[ \text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder} \]In our exercise, after performing polynomial long division, we obtain the quotient as \( q(x) = 2x + \frac{5}{3} + \frac{1}{3} \) and the remainder as \( r(x) = -9 \). This shows how the initial polynomial is broken down into simpler parts.

Division of Polynomials

Dividing polynomials can often seem daunting, but it follows a process similar to dividing whole numbers. The division of polynomials involves several steps, which need to be repeated until no further division is possible. Here’s how it works:

• Set up the division: Arrange your divisor and dividend as you would in regular long division, with the dividend inside the division bracket.
• Begin the division: Divide the leading term of the dividend by the leading term of the divisor to find the first term of your quotient.
• Multiply and subtract: Multiply the entire divisor by the term you just found and subtract this from your dividend. This gives you a new, smaller polynomial to work with.
• Repeat: Continue this process. Divide the new dividend’s leading term by the divisor's leading term, multiply, and subtract until the degree of the new dividend is less than the degree of the divisor.

By following these steps, you systematically break down the polynomial into simpler components, allowing you to identify the quotient and remainder. Our exercise exemplifies this technique, where each step contributes to simplifying a complex expression into its basic elements.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. Polynomials are a particular class of algebraic expressions that can range from simple monomials to complex expressions involving several terms. Understanding how to manipulate these expressions through operations like addition, subtraction, multiplication, and division is crucial in algebra. The division of polynomials we performed is a core technique used to simplify algebraic expressions.

• Coefficient: The numerical part of a term. In the expression \(6x^3\), 6 is the coefficient.
• Variable: The letter, often \(x\), that stands for an unknown value.
• Exponent: This indicates how many times a variable is multiplied by itself. In \(x^3\), the exponent is 3.

Through polynomial long division, you see how algebraic expressions can be systematically broken down, making it easier to understand and solve more complex equations. It's about recognizing patterns and following a step-by-step process to simplify what looks complex at first glance.