Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Use long division to divide. Divisor \(x+3\) Dividend $$2 x^{2}+10 x+12$$

Step-by-Step Solution

Answer

By dividing \( 2x^2 + 10x + 12 \) by \( x+3 \), the result is \(2x + 4\). There is no remainder.

Step 1: Setup the Division

Set up the long division similarly to how numerical long division is set. Write the dividend \(2x^2 + 10x + 12 \) inside the division symbol and the divisor \( x+3 \) outside the division symbol.

Step 2: Divide the Leading Terms

Divide the first term of the dividend, \( 2x^2 \), by the first term of the divisor, \( x \). The result is \( 2x \), which is written on top of the division symbol.

Step 3: Multiply Back and Subtract

Multiply the divisor \( x+3 \) by \( 2x \) from the result of the division above, giving \( 2x^2 + 6x \). Write this under the dividend and proceed to subtract \( 2x^2 + 6x \) from \( 2x^2 + 10x + 12 \).

Step 4: Bring Down the Next Term

After subtracting, bring down the next term from the dividend. Here you'll have \( 4x+12 \) in this case.

Step 5: Continue the Division

Repeat the process for the next term. Divide the new first term of the dividend \( 4x \) by the first term of the divisor \( x \), to yield \( 4 \). Write this result above the division symbol, next to \( 2x \). Then multiply back, subtract and check if there are still terms from the dividend to bring down.

Step 6: Repeat until Remains can't be Divided

Continue the process until all terms from the dividend are used up or what remains of the dividend after subtraction can't be divided by the divisor anymore. What remains is the remainder of the division.

Step 7: Write the Final Result

The final result contains the quotient and the remainder (if any). Write down the solution in the format Quotient + Remainder/Divisor.

Key Concepts

Long Division in Algebra

In algebra, long division is an extension of the technique used in arithmetic, meant for dividing polynomials. Just like with numbers, you arrange the terms from the highest degree to the lowest. This setup helps keep the process organized and accurate.

Here's how long division works in algebra:

• Setup: Begin by placing the dividend inside the division symbol and the divisor outside. Arrange both polynomial expressions in descending order of powers.
• Division of Leading Terms: Identify the leading term of the dividend and divide it by the leading term of the divisor. This gives the first term of the quotient.
• Multiplication and Subtraction: Multiply the entire divisor by the term obtained in the previous step. Place this result under the corresponding terms of the dividend, and subtract these terms from the dividend.
• Bringing Down the Next Term: If there are any remaining terms in the dividend, bring down the next term and repeat the division process.
• Repeat: Continue this sequence until all terms have been addressed. What’s left is the remainder, if any.

Long division in algebra may seem daunting at first, but with practice, you will find it a systematic and straightforward process. It is an essential skill for any student challenged with polynomial problems.

Polynomials

Polynomials are expressions composed of variables, coefficients, and exponents. These are combined using addition, subtraction, and multiplication. Each part of a polynomial is called a "term," and they are classified by the degree, which is the highest exponent of the variable.

Characteristics of polynomials include:

• Terms: A term includes a constant multiplied by a variable raised to an exponent. For example, in the term \(2x^2\), 2 is the coefficient, \(x\) is the variable, and 2 is the exponent.
• Degree: The degree of a polynomial is determined by the term with the highest exponent. For example, \(2x^2 + 10x + 12\) is a polynomial of degree 2.
• Types: Polynomials can be categorized by the number of terms: monomials, binomials, trinomials, etc.

Understanding the makeup of polynomials is necessary for performing operations such as addition, subtraction, multiplication, and division, especially when delving into algebraic expressions.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and operations like addition, subtraction, multiplication, and division. Unlike equations, expressions do not have an equality sign. Algebraic expressions are used to describe mathematical relationships in a generalized form.

Some important points to remember about algebraic expressions include:

• Components: They consist of terms, with each term being a product of numbers and variables.
• Variables: Represent unknown quantities and can change in value.
• Constants: Are fixed values within an expression.
• Simplification: Expressions can be simplified by combining like terms and using operations effectively.

Recognizing and simplifying algebraic expressions is a fundamental part of algebra, aiding in solving equations and inequalities by isolating variables and constants.