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 is \(q(x) = 2x - 1\) and the remainder is \(r(x) = -3x + 2\).

Step 1: Set Up Long Division

To begin, set up the long division similar to how it is usually done with numbers. Put \(6x^3 + 13x^2 - 11x - 15\) inside the division symbol and \(3x^2 - x - 3\) on the outside.

Step 2: Division

The first term of the divisor, 3x^2, divides the first term of the dividend, 6x^3, to give \(2x\), which is the first term of the quotient. Multiply the divisor, \(3x^2 - x - 3\), by \(2x\) and subtract the result from the dividend.

Step 3: Repeat

Repeat step 2 with the new dividend, which is now the remainder of the previous subtraction. Divide the first term of the divisor into the first term of the new dividend, multiply the divisor by this new part of the quotient, and subtract again.

Step 4: Continue until not possible

Continue to repeat step 3 until it is no longer possible to subtract because the degree of the remainder is less than the degree of the divisor. The quotient, \(q(x)\), is the result of each division step, and \(r(x)\), the remainder, is the result of the last subtraction.

Key Concepts

Quotient

When we perform polynomial long division, the quotient is the central figure. Think of the quotient as the main result of the division. In simpler terms, it's what you get when you "divide" the dividend by the divisor.
In this context, the dividend is the polynomial you are dividing, and the divisor is the polynomial you are using to divide. For the given exercise:

• Dividend: \(6x^3 + 13x^2 - 11x - 15\)
• Divisor: \(3x^2 - x - 3\)

As you proceed with polynomial division, you find pieces of the quotient bit by bit. For each term of the quotient, divide the first term of the current dividend by the first term of the divisor.
This approach is similar to long division with numbers, but involves variables and keeping track of powers (also known as degrees) of those variables. The quotient lets you know how many times the divisor can fit into the dividend, expressed as a polynomial.

Remainder

In polynomial division, just like with number division, there might be a leftover part which cannot be divided further due to its exponent being smaller than that of the divisor. This leftover part is known as the remainder.
The remainder in polynomial division is crucial. It tells you what is left after the division process has been completed as far as it can go. In the exercise provided, after each stage of division and the subtraction process, the remainder becomes the new dividend.
Once the degree of the remainder polynomial is less than the degree of the divisor polynomial, the division process stops. The remainder then could be a polynomial of a lower degree or even a zero polynomial, indicating that the dividend is completely divisible by the divisor without any leftover.

Polynomial Division

Polynomial division is a process similar to conventional numerical division but involves polynomials. Polynomials are algebraic expressions comprising coefficients, variables, and exponents.
This method of division requires subtracting multiples of a polynomial, minimizing the dividend until the end of the process. The goal is to find both the quotient and remainder.
Here's a quick breakdown of the steps involved:

• Align polynomials: Arrange the dividend and divisor in descending powers of the variable.
• Divide the leading terms: Divide the first term of the dividend by the first term of the divisor to determine the first term of the quotient.
• Multiply and subtract: Multiply the entire divisor by the resultant term of the quotient and subtract from the dividend.
• Repeat: Use the new polynomial from subtraction as the new dividend and repeat the process until possible.

This method requires careful attention to the degree of variables and ensuring each subtraction accurately reflects the multiplication of polynomials.

Degree of Polynomial

The degree of a polynomial is one of its most important features. It represents the highest power of the variable present in the polynomial. Understanding the degree helps in assessing how the polynomial will interact during division.
In polynomial long division, the degree of the polynomial plays a crucial role in determining when to stop the division process. The process continues until the degree of the remainder is less than the degree of the divisor.
For the given exercise:

• The degree of the dividend \(6x^3 + 13x^2 - 11x - 15\) is 3, owing to the highest term \(6x^3\).
• The degree of the divisor \(3x^2 - x - 3\) is 2, with \(3x^2\) as the highest term.

The degree essentially acts as a stopping point and lets you know that further division can't be effectively executed, which results in what becomes the remainder of the division.