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Chapter 6: Problem 12

For Problems \(11-46\), factor each polynomial completely. (Objective 2) $$ 18 x+24 y $$

Short Answer

Expert verified
The polynomial factors completely to \(6(3x + 4y)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

The expression given is \(18x + 24y\). The first step in factoring is to find the greatest common factor of the coefficients of the terms involved, which are 18 and 24. The greatest common factor of 18 and 24 is 6.
02

Factor Out the GCF

Once the GCF is identified (which is 6), we factor it out from each term in the polynomial. For the term \(18x\), when divided by 6, the result is \(3x\). For the term \(24y\), when divided by 6, the result is \(4y\). Therefore, factoring 6 out of the polynomial gives \(6(3x + 4y)\).
03

Verify the Factorization

To ensure that the factorization is correct, distribute 6 into \(3x + 4y\) to see if you obtain the original expression. Calculating this gives us \(6 \cdot 3x + 6 \cdot 4y = 18x + 24y\), which matches the original expression, confirming the factorization is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Factor
When factoring polynomials, one of the first steps is to identify the Greatest Common Factor (GCF). The GCF is the largest number that can evenly divide each of the polynomial's terms. It is important because it simplifies expressions and makes solving equations easier.

Finding the GCF involves looking at the coefficients of the terms in the polynomial. In the expression \(18x + 24y\), we need to find the GCF of 18 and 24.
  • First, list out the factors of each number:
    • The factors of 18 are 1, 2, 3, 6, 9, and 18.
    • The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.
  • Next, identify the largest factor that appears in both lists. This is 6.
The GCF is a foundational step to simplify many algebraic processes and is essential when rewriting expressions in a more usable form.
Binomial Expression
A binomial expression is a type of polynomial that contains exactly two terms. In the context of our problem, the original expression \(18x + 24y\) is a binomial because it contains two terms connected by a plus sign. Understanding how to work with binomials is vital, as they frequently appear in algebra problems.

Each term in a binomial consists of a numerical coefficient and a variable part. Here:
  • The first term is \(18x\), where 18 is the coefficient and \(x\) is the variable.
  • The second term is \(24y\), where 24 is the coefficient and \(y\) is the variable.
In many algebraic processes, especially factorization, managing binomials by identifying common factors allows for significant simplification. This not only aids in current problem solving but prepares you for more complex expressions, which may result from or be reduced into binomials.
Polynomial Factorization
Polynomial factorization is a mathematical process used to express a polynomial as a product of its factors. It's similar to breaking down a composite number into its prime factors but applied to algebraic expressions.

Consider the expression \(18x + 24y\) from our exercise:
  • We started by identifying the GCF, which was 6, from the coefficients of the polynomial.
  • We then "factored out" this 6, meaning we divided each term by the GCF to simplify the polynomial into \(6(3x + 4y)\).
  • Factoring checks involve redistributing the GCF back through the binomial to ensure the original polynomial is obtained, confirming the accuracy of the factorization.
Factorization can reveal hidden structures within polynomials, making equations easier to solve or expressions further reducible. It is an essential skill for any student progressing in algebra.

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