Problem 95 - Elementary Algebra

Question

(a) Describe the mistake in words, or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: Factor: \(\left(-6 x^{2}+4 x z\right)\) Incorrect Answer: \(\left(-6 x^{2}+4 x z\right)\) $$ =-2 x(3 x+2 z) $$

Step-by-Step Solution

Answer

-6x^2 + 4xz should be factored as 2x(-3x + 2z).

Step 1 - Identify the mistake

The mistake is in factoring the expression \(-6x^2 + 4xz\). Specifically, the greatest common factor (GCF) has not been applied correctly. The incorrect answer provided is \(-2x(3x + 2z)\), but this incorrectly factors the terms.

Step 2 - Find the GCF

Identify the GCF of the terms \(-6x^2\) and \(+4xz\). The GCF of \(-6x^2\) and \+4xz\ is \2x\, not \-2x\.

Step 3 - Factor out the GCF

Factor \2x\ out of each term in the expression. \(-6x^2\) divided by \2x\ gives \-3x\.\(4xz\) divided by \2x\ gives \+2z\. So, the correct factorization should be: \2x(-3x + 2z)\.

Step 4 - Write the correct solution

Write the fully factored form of the original expression as: \(2x(-3x + 2z)\).

Key Concepts

greatest common factor (GCF)

When factoring algebraic expressions, identifying the Greatest Common Factor (GCF) is a crucial first step.
The GCF of an expression is the largest factor that can evenly divide each term within the algebraic expression.
For example, in the expression oway-6x^2 + 4xzoh, each term contains an 'x' and the coefficients are -6 and 4, which share a common factor of 2.
This means the GCF for this expression is 2x, not -2x.

• Finding the GCF helps in simplifying expressions more accurately.
• It is important to always check each coefficient and variable for common factors.
• Misidentifying the GCF leads to incorrect factorizations, as seen in the problem's incorrect solution:

The initial expression: ewline -6x^2 + 4xzewline Factored incorrectly as:ewline -2x(3x + 2z)ewline Correct factorization using GCF (2x) is:ewline 2x(-3x + 2z)ewline Understanding GCF ensures accuracy in broader algebraic operations.

factorization steps

Factoring expressions is a step-by-step method.
Sticking to these steps ensures a correct and simple result. Let's go over the true factorization process:

• Step 1: Identify the Greatest Common Factor (GCF) - Find the largest term or coefficient that is common to each term in the expression. For instance, in ewline -6x^2 + 4xz, the GCF is 2x.

• Step 2: Divide each term by the GCF - When each term is divided by the GCF, the coefficients and variables simplify. For instance, ewline -6x^2 ÷ 2x = -3x and 4xz ÷ 2x = 2z.

• Step 3: Write the simplified terms as a product with the GCF - Combine the results into a new expression where the GCF is factored out. This means -6x^2 + 4xz turns into 2x(-3x + 2z).

By meticulously following these steps, you avoid mistakes and accurately factor the expression.

algebraic mistakes

In algebra, small errors can lead to significant incorrect outcomes.
It's essential to develop a habit of checking your work and identifying common mistakes.
Some typical mistakes include:

• Incorrect GCF identification: As seen in the problem, misidentifying the GCF (-2x instead of the correct 2x) leads to an incorrect factorization.
• Sign errors: Factor sign mistakes are common and need close attention. Verify each term's sign during and after factorization.
• Omission of terms: Sometimes, terms can be mistakenly left out during factorization. Correctly factor all terms to ensure nothing is missing.

Detecting these mistakes early helps in developing better skills. Practice always checking every step:

• Check each term for GCF precision.
• Double-check all signs (negative and positive).
• Ensure all terms appear in the final factored expression.

A solid grasp of these concepts builds stronger algebra skills and better problem-solving strategies.