Problem 10 - College Algebra with Modeling and Visualization

Question

Factor out the greatest common factor:. \(14 a^{4}-21 a^{2}+35 a\)

Step-by-Step Solution

Answer

The greatest common factor is \(7a\); the expression factors to \(7a(2a^3 - 3a + 5)\).

Step 1: Identify the Terms

The given expression is \(14a^4 - 21a^2 + 35a\). We need to identify each term in the expression: \(14a^4\), \(-21a^2\), and \(35a\).

Step 2: Find the GCF of the Coefficients

Next, we find the greatest common factor (GCF) of the coefficients, which are 14, 21, and 35. The GCF of these numbers is 7.

Step 3: Find the GCF of the Variables

The variable terms are \(a^4\), \(a^2\), and \(a\). The lowest power of \(a\) present in all terms is \(a\). Therefore, the GCF for the variables is \(a\).

Step 4: Combine the GCFs

Now, combine the GCF of the coefficients, 7, and the GCF of the variables, \(a\), to get the overall GCF of the expression, which is \(7a\).

Step 5: Factor Out the GCF

Factor \(7a\) out of each term in the expression: \(14a^4 - 21a^2 + 35a = 7a(2a^3) - 7a(3a) + 7a(5)\).

Step 6: Simplify the Expression

Simplify the factored expression: \(7a(2a^3 - 3a + 5)\).

Step 7: Verify the Solution

Distribute \(7a\) back through the expression to ensure accuracy: \(7a(2a^3) = 14a^4\), \(7a(-3a) = -21a^2\), and \(7a(5) = 35a\). This confirms the original expression is \(14a^4 - 21a^2 + 35a\).

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor, often abbreviated as GCF, is a fundamental concept in algebra that helps in simplifying expressions. A GCF is the largest factor that divides two or more numbers or terms without leaving a remainder. Finding the GCF is a crucial first step in factoring algebraic expressions.

Let's break down the steps necessary to find the GCF for an expression like the one given in the exercise:

• **Identify the Coefficients and Variables**: Look at all the numbers and variables in each term of the expression. For example, in the expression \(14a^4 - 21a^2 + 35a\), the coefficients are 14, 21, and 35, and the variable present in all terms is \(a\).
• **Find the GCF of the Coefficients**: Determine the highest number that divides each of the coefficients. In this case, the GCF of 14, 21, and 35 is 7.
• **Find the GCF of the Variables**: Identify the lowest power of the variable that is common across all terms. Here, the smallest power of \(a\) in all terms is \(a^1\). Therefore, the GCF for the variable part is \(a\).
• **Combine These GCFs**: Multiply the GCF of the coefficients by the GCF of the variables. In this example, this gives us a GCF of \(7a\).

With the GCF identified, you can then factor it out of the entire expression to simplify it.

Polynomials

Polynomials are an essential area of algebra involving expressions that consist of variables and coefficients, arranged in terms of powers. A typical polynomial might look like \(14a^4 - 21a^2 + 35a\).

Here are some key features of polynomials:

• **Terms**: Each part of a polynomial, separated by a plus or minus sign, is called a term. In our example, the terms are \(14a^4\), \(-21a^2\), and \(35a\).
• **Degrees**: The degree of a polynomial is the highest power of the variable present. Here, the degree is 4 because of the term \(14a^4\).
• **Monomials, Binomials, and Trinomials**: A polynomial with one term is a monomial, two terms is a binomial, and three terms is a trinomial, which is our case here.

Understanding polynomials is crucial because they are used to model situations and solve equations in various applications. Factoring polynomials, such as removing the GCF from each term, makes them easier to handle and solve in equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are fundamental building blocks in algebra and crucial for representing mathematical ideas.

Here are a few important aspects to consider:

• **Variables**: These are symbols like \(a\) used to represent numbers in expressions. Variables allow us to create general expressions that work for many different numbers.
• **Operations**: The fundamental operations used in algebraic expressions include addition, subtraction, multiplication, and division. The given expression uses these operations to combine terms.
• **Simplification**: One of the main goals when working with algebraic expressions is to simplify them. This can mean factoring, as in our exercise, combining like terms, or performing arithmetic operations to make the expression as concise as possible.

Mastering algebraic expressions is important for solving equations and understanding relationships in mathematics. Simplification by factoring not only simplifies the expression itself but also makes solving equations more manageable.