Problem 13 - College Algebra with Corequisite Support

Question

For the following exercises, use long division to divide. Specify the quotient and the remainder. $$ \left(2 x^{3}+3 x^{2}-4 x+15\right) \div(x+3) $$

Step-by-Step Solution

Answer

Quotient: \(2x^2 - 3x + 5\); Remainder: \(0\).

Step 1: Set Up the Division

Write down the long division format. Place the dividend \(2x^3 + 3x^2 - 4x + 15\) inside the division symbol and the divisor \(x + 3\) outside.

Step 2: Divide the Leading Terms

Divide the leading term of the dividend \(2x^3\) by the leading term of the divisor \(x\), which gives \(2x^2\). This quotient term is written above the division line.

Step 3: Multiply and Subtract

Multiply \(2x^2\) by the divisor \(x + 3\), resulting in \(2x^3 + 6x^2\). Subtract this from the original dividend to find the new dividend \(-3x^2 - 4x\).

Step 4: Repeat the Process

Divide the new leading term \(-3x^2\) by \(x\), which gives \(-3x\). Multiply \(-3x\) by \(x + 3\) to get \(-3x^2 - 9x\). Subtract this from the current dividend to get \(5x + 15\).

Step 5: Final Round of Division

Divide \(5x\) by \(x\) to obtain \(5\). Multiply \(5\) by \(x + 3\), resulting in \(5x + 15\). Subtract \(5x + 15\) from the current dividend getting a remainder of \(0\).

Step 6: Write Down the Quotient and Remainder

After completing the division, the quotient is \(2x^2 - 3x + 5\) and there is no remainder, \(0\).

Key Concepts

Long Division in Algebra

Long division in algebra might sound complicated at first, but it's an extremely useful method for dividing polynomials. It's similar to the long division you may have learned with numbers. However, instead of numbers, we work with terms of polynomials. This involves taking each term of the polynomial one at a time and performing operations.

When you set up long division, you write your polynomial (the dividend) under the long division symbol. The divisor, another polynomial, is placed outside. In our example, the polynomial is placed as follows:

• The dividend is: \(2x^3 + 3x^2 - 4x + 15\)
• The divisor is: \(x + 3\)

The method involves dividing, multiplying, and subtracting sequentially. It requires attention to the highest power term of the dividend and the highest power term of the divisor, often referred to as the "leading terms." By understanding long division in algebra, students can divide polynomials without the heavy reliance on polynomial calculators.

Quotient and Remainder

In the context of polynomial division, the quotient and remainder play an important role. After performing long division, the quotient is the result you obtain above the division bar, while the remainder is what's left after all the possible divisions are done.

In numeric division, you might recall how you would divide, sometimes ending up with a remainder that represents what couldn't be further divided by the divisor. Similarly, in polynomial division, if our remainder isn't zero, it's simply another polynomial term that can't be divided further by the given divisor.

For the example we solved, our calculations ended with:

• Quotient: \(2x^2 - 3x + 5\)
• Remainder: \(0\)

Here, since the remainder is zero, the divisor perfectly divides the dividend, indicating a "clean" division. Understanding the meaning of these terms is crucial, as they often contribute to solving broader problems involving polynomial functions.

Polynomial Functions

Polynomial functions are expressions that consist of variables raised to whole number exponents, combined using addition or subtraction. They are incredibly versatile and appear frequently in algebra.

A polynomial function is typically written in descending order of the power of its variables, a form you might recognize from expressions like \(a_n x^n + a_{n-1} x^{n-1} + \, ... \, + a_1 x + a_0\). Here, each \(a_i\) represents a coefficient, which can be any real number.

• The polynomial \(2x^3 + 3x^2 - 4x + 15\) is an example where:
• "\(2x^3\)" signifies a term with a coefficient of 2 and an exponent of 3.
• The power of the variable determines the degree of the polynomial, which is 3 in this instance.

Understanding polynomial functions allows you to do more than just perform operations like division. They're essential in graphing equations, calculating slopes, finding maximum and minimum values of functions, and modeling real-world situations. Mastery over these functions creates a solid foundation for tackling more complex algebraic situations.