Problem 14 - Elementary Algebra

Question

For the following problems, factor the polynomials. $$ 16 f-36 $$

Step-by-Step Solution

Answer

Answer: The factored form of the expression is 4(4f-9).

Step 1: Identify the GCF

To begin factoring, find the greatest common factor of 16f and -36. In this case, the GCF is 4.

Step 2: Factor out the GCF from each term

Divide each term by the GCF, which is 4, and write the result multiplied by the GCF: $$ 4(4f - 9) $$

Step 3: Confirm the result

Make sure the factored expression is correct by distributing the GCF back across the terms. If the result matches the original expression, the solution is correct: $$ 4(4f - 9) = 16f - 36 $$ The factored expression is $$4(4f-9)$$.

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF), also known as the greatest common divisor, is a fundamental skill in algebra. It's the largest number that divides two or more numbers without any remainder. To factor a polynomial, finding the GCF of its terms is the first step.

For instance, consider the polynomial 16f - 36. Both terms, 16 and -36, are divisible by 4. So, the GCF here is 4. By identifying and factoring out the GCF, we can simplify expressions, making them easier to handle in subsequent algebraic processes.

Moreover, knowing how to find the GCF paves the way for solving more complex algebraic problems such as solving equations or simplifying fractions. It's the cornerstone of breaking down expressions into their simplest form, an essential aspect of understanding algebra.

Algebraic Expressions

An Algebraic Expression is a combination of variables, numbers, and operations. Variables represent unknown values and are generally denoted by letters, such as 'f' in our example polynomial 16f - 36.

An important aspect of algebraic expressions is their versatility; they can model real-world situations and solve a variety of problems. When we factor expressions, like finding 4(4f - 9) as the factored form of 16f - 36, we are essentially breaking them down into simpler, more manageable pieces.

Factoring an algebraic expression enables students to understand and solve complex equations more easily. Recognizing and manipulating these expressions are of paramount importance in high school math and beyond, laying the groundwork for future studies in mathematics and science.

Distributive Property

The Distributive Property is a keystone in algebra that allows us to multiply a single term by each term inside a parenthesis. Essentially, it involves spreading out or distributing the multiplication over addition or subtraction.

For example, by applying the distributive property to 4(4f - 9), we multiply 4 by each term inside the parentheses, yielding 16f - 36—the original polynomial we started with. This demonstrates that our factoring was done correctly.

Understanding this property is crucial when it comes to simplifying expressions, solving equations, and factoring polynomials. It's also integral in proving that the factored form of an expression is equivalent to the original. Employing the distributive property effectively can transform seemingly intricate problems into more straightforward ones that are easier to manage.