Problem 6 - College Algebra and Calculus: An Applied Approach

Question

V\(3 x(x+2)-4(x+2)\)

Step-by-Step Solution

Answer

The solution to this problem is \(3x^2+2x-8\).

Step 1: Distribute First

Use the distributive property to multiply through the parentheses: V\(3x^2+6x-4x-8\)

Step 2: Combine Terms

Next, combine like terms: V\(3x^2+2x-8\)

Key Concepts

Combining Like Terms

When working with algebraic expressions, combining like terms is a crucial skill. Like terms are terms in an expression that have the same variables raised to the same powers. They can be combined by adding or subtracting their coefficients. Consider the expression:

• V\(3x^2 + 6x - 4x - 8\)

Here, the like terms are found in the linear terms \(6x\) and \(-4x\). Each of these terms contains the variable \(x\). So, they can be combined:

• Add the coefficients of the like terms: \(6x - 4x = 2x\)

After combining, our expression becomes:

• V\(3x^2 + 2x - 8\).

By simplifying an expression to combine like terms, you make it more manageable and easier to solve or further manipulate.

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and coefficients, involving the operations of addition, subtraction, and multiplication. They are often written in descending order of the power of their variable. For instance, consider the expression V\(3x^2 + 2x - 8\), which is a polynomial. This expression includes:

• A quadratic term: \(3x^2\), where 2 is the degree of the term
• A linear term: \(2x\), where 1 is the degree of the term
• A constant: \(-8\), which doesn't have a variable

Each term in a polynomial is separated by addition or subtraction. The entire expression is termed as a trinomial because it contains three terms.

Factoring

Factoring is the process of breaking down an expression into multiples, known as factors, which can be multiplied together to obtain the original expression. It is a vital step when solving polynomial equations, making them simpler. For the polynomial V\(3x^2 + 2x - 8\), factoring can help to identify solutions to polynomial equations or simplify expressions for further calculations. To factor this polynomial, you look for common factors in each term, or recognize it as a binomial or trinomial pattern that can be broken down, such as using techniques like:

• Finding greatest common factors (GCF)
• Applying methods like trial and error for trinomial factoring, and
• Using special factoring formulas like difference of squares or perfect square trinomials

While the process can vary depending on the specific polynomial, mastering factoring techniques facilitates solving complex algebraic problems.