Question

\(6 a^{2}+8 a b-4 a c\) is equivalent to which of the following expressions? (A) \(a(3 a+4 b+2 c)\) (B) \(2 a(3 a+4 b-2 c)\) (C) \(4 a(a+b-2 c)\) (D) \(2 a(3 a-4 b+2 c)\)

Short answer

The correct answer is (B) \(2a(3a + 4b - 2c)\).

Step-by-step solution

Identify the Common Factor

The given expression is \(6a^2 + 8ab - 4ac\). To factor it, first look for the greatest common factor (GCF) among all the terms. Observing each term, we see that all terms have the factor of \(2a\). So, \(2a\) can be factored out from the expression.

Factor Out the Common Factor

Using the common factor identified in Step 1, we can write:\[6a^2 + 8ab - 4ac = 2a(3a + 4b - 2c)\]Here, each term in \(6a^2 + 8ab - 4ac\) has been divided by \(2a\), resulting in \(3a + 4b - 2c\).

Match the Expression to Options

From Step 2, the factored expression is \(2a(3a + 4b - 2c)\). Now, compare it with the options given:- (A) \(a(3a + 4b + 2c)\)- (B) \(2a(3a + 4b - 2c)\)- (C) \(4a(a + b - 2c)\)- (D) \(2a(3a - 4b + 2c)\)The correct match is option (B) \(2a(3a + 4b - 2c)\).

Key concepts

Factoring Polynomials

When you hear the term "factoring polynomials," it might sound like a mouthful, but it's actually a helpful trick in algebra. Factoring a polynomial is like breaking it down into smaller pieces, much like breaking down a big block of chocolate into smaller, delicious squares! This is particularly useful because it makes dealing with polynomials much easier.

To factor a polynomial, you need to look for common components in each term of the expression. In our exercise, the polynomial is \(6a^2 + 8ab - 4ac\), and our goal was to factor this into simpler components. By identifying the common factor across all terms, in this case, \(2a\), we can "pull out" this factor, breaking the polynomial into \[2a(3a + 4b - 2c)\].

This means that each of the terms inside matches an original term but is now expressed in a different form. Factoring can help solve equations more easily, simplify expressions, and even solve real-world problems. So next time you see a complex polynomial, remember that factoring is there to make your life easier!

Greatest Common Factor

The Greatest Common Factor (GCF) is a key player in simplifying algebraic expressions. Think of it as the common thread that runs through some numbers or terms. The GCF is the largest value that evenly divides each of the terms in an expression.

In our example, we have the expression \(6a^2 + 8ab - 4ac\). When we searched for a common factor across all terms, we discovered that \(2a\) can divide \(6a^2\), \(8ab\), and \(-4ac\) without leaving a remainder. Thus, \(2a\) is the greatest common factor.

• For \(6a^2\), \(2a\) divides to give \(3a\).
• For \(8ab\), \(2a\) divides to give \(4b\).
• For \(-4ac\), \(2a\) divides to give \(-2c\).

Factoring out the GCF simplifies our expression to \[2a(3a + 4b - 2c)\]. Recognizing the GCF helps streamline expressions, making challenging algebra problems much more manageable. This basic approach of finding and using the GCF can be applied over and over in algebra and beyond!

Mathematical Problem-Solving

Solving mathematical problems, especially in algebra, often involves a series of logical steps, and knowing these steps can turn a problem from puzzling to plain. Our original challenge was to see which expression \(6a^2 + 8ab - 4ac\) matched, after factoring. The logical flow in solving this was like putting together a puzzle.

Here's the approach we took:

• First, identify the problem: Recognize that we're given a polynomial and some options to match.
• Next, gather the tools: Determine the greatest common factor, which is \(2a\).
• Then, factor the expression: Simplify \(6a^2 + 8ab - 4ac\) using that GCF to get \[2a(3a + 4b - 2c)\].
• Finally, compare to options: Match our simplified expression to those given. This brings us to option (B), which is the correct solution.

By breaking down the problem into these clear steps, we transition from an overwhelming problem to a more enjoyable solving process. This isn’t just helpful in algebra but in any logical problem-solving you may face - identifying known elements, simplification, and finally matching solutions. Approach every problem methodically, and it will seem less daunting!