Problem 9 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 8 b+12 $$

Step-by-Step Solution

Answer

Question: Factor the polynomial 8b + 12. Answer: The factored form of the polynomial 8b + 12 is 4(2b + 3).

Step 1: Identify the GCF of the coefficients

To get started, we need to find the GCF of the coefficients, which are 8 and 12. We can do this by listing their factors and choosing the largest factor that both numbers have in common. The factors of 8 are 1, 2, 4, and 8; the factors of 12 are 1, 2, 3, 4, 6, and 12. We can see that the largest common factor is 4.

Step 2: Factor the GCF out of the polynomial

Next, we will factor the GCF (4) out of the polynomial. This means we will divide each coefficient by 4 and write the polynomial as a product with 4 as one of the factors. $$ 8b + 12 = 4(2b + 3) $$

Step 3: Check the factored polynomial

Finally, it's always a good idea to double-check our work. We can do this by expanding the factored polynomial to see if we get the original polynomial back. $$ 4(2b + 3) = 4(2b) + 4(3) = 8b + 12 $$ Our factored polynomial is correct, so the final answer is: $$ 8b + 12= 4(2b + 3) $$

Key Concepts

Greatest Common Factor (GCF)

The greatest common factor (GCF) is a fundamental concept when simplifying expressions, including inequalities and polynomials. It refers to the largest number that can divide each term of your given set without leaving a remainder.
Finding the GCF involves a straightforward process. You list the factors of each number involved and identify the greatest number common to those factors. For the numbers 8 and 12, as in our exercise, the factors are:

• Factors of 8: 1, 2, 4, 8
• Factors of 12: 1, 2, 3, 4, 6, 12

Among these, the greatest common factor is 4. Once the GCF is identified, it can be factored out from the polynomial, simplifying the expression and making it easier to work with.
Remember, the GCF is not only important for simplifying expressions but also plays a significant role in solving polynomial equations and understanding their behavior. Try practicing finding the GCF of different sets of numbers to get comfortable with this essential algebraic skill.

Polynomial Equations

Polynomial equations are equations involving polynomials, which are expressions composed of variables and coefficients, connected by addition, subtraction, and multiplication. Polynomials can consist of multiple terms.
Solving these equations often involves factoring, and the GCF is vital because we use it to simplify these polynomials, making them easier to manipulate and solve.
For instance, let's take the polynomial equation given in our exercise:

• Original Polynomial: \(8b + 12\)

To solve the equation, we start with simplifying it using the GCF identified as 4. Thus, factoring the equation, we rewrite it as\(4(2b + 3)\).
Factoring isn't just for simplifying; it also helps us find solutions. Whether the polynomial is set equal to zero or another number, recognizing the factors sets the groundwork for finding roots or solutions of the polynomial. Understanding how to manipulate and solve polynomial equations is key in algebra and more advanced mathematics.

Algebraic Expressions

An algebraic expression is a mathematical phrase combining numbers, variables, and operations. These expressions do not include equality signs. Understanding how to work with algebraic expressions is foundational for anyone studying algebra.
In the given example, \(8b + 12\) is an algebraic expression. By learning to identify the GCF, we can effectively factor these expressions, simplifying them for further use.

• Before Factoring: \(8b + 12\)
• After Factoring: \(4(2b + 3)\)

The process of factoring involves dividing the whole expression by the GCF, restructuring it into a more workable form. Once it's factored, the expression can reveal more information which can be useful in broader algebraic equations or systems.
Algebraic expressions form the building blocks of algebra, and understanding how to manipulate them using principles like factoring allows for enhanced problem-solving capabilities.