Problem 8 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 6 a+24 $$

Step-by-Step Solution

Answer

Answer: The factored form of the polynomial \(6a + 24\) is \(6(a + 4)\).

Step 1: Find the greatest common factor (GCF)

To find the greatest common factor of the terms in the polynomial, we need to consider the coefficients and variable factors of each term. The terms in the polynomial are \(6a\) and \(24\). The greatest common factor of their coefficients is 6.

Step 2: Factor out the GCF

Now that we have found the GCF (6), we need to factor it out of the polynomial. To do this, we divide each term in the polynomial by the GCF and then multiply the result by the GCF, like this: $$ 6a + 24 = 6(a + 4) $$

Step 3: Write the factored polynomial

The factored form of the polynomial is the result obtained by factoring out the GCF. In this case, the factored form of the polynomial is: $$ 6(a + 4) $$

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial concept in algebra, especially when dealing with polynomials. To find the GCF of terms in a polynomial, consider both the coefficients (the numbers in front of variables) and any variables they may have in common.

• Look at each term in the polynomial separately.
• Identify the highest number that evenly divides each of these coefficients. This number is your coefficient GCF.
• If there are any variables present, determine the highest power of common variables in each term.

For example, in the polynomial \(6a + 24\), the GCF is 6, since 6 is the largest number that divides both 6 and 24 evenly. Understanding the GCF helps simplify expressions, making polynomial problems easier to handle.

Polynomial Factoring

Factoring polynomials is a method used in algebra to simplify an algebraic expression or solve polynomial equations. It involves breaking down a complex polynomial into simpler factors that when multiplied together give back the original polynomial.
Here's a simple approach to factor a polynomial like \(6a + 24\):

• Identify the GCF first, which we've found to be 6.
• Divide each term of the polynomial by the GCF.
• This gives a new, simpler expression inside parentheses, e.g., \( (a + 4) \).
• Multiply the factored expression by the GCF to verify the factoring is correct.

The goal of factoring is often to make complex equations manageable or find potential variables' values that make the polynomial equal to zero, which are known as roots.

Algebraic Expressions

Algebraic expressions form the core framework of algebra, which involves numbers, variables, and the arithmetic operations like addition and multiplication. A polynomial is a type of algebraic expression that includes terms which can be constants, variables, or a product of both.
Understanding algebraic expressions starts with recognizing their parts:

• **Terms**: Components that are added together, such as \(6a\) and \(24\) in our example.
• **Coefficients**: The numerical part of a term, for instance, 6 in \(6a\).
• **Variables**: Symbols used to represent unknowns, such as \(a\).

Learning how to manipulate these expressions, such as factoring them, is essential for solving equations and understanding more complex algebraic concepts.