Problem 40 - Algebra 1: Concepts and Skills

Question

Factor the expression completely. \(-12 z^{3}+3 z^{2}\)

Step-by-Step Solution

Answer

The fully factored expression of \( -12 z^{3} + 3 z^{2} \) is \(3z^{2}( -4 z + 1 )\).

Step 1: Identify the Greatest Common Factor (GCF)

Analyze the expression \( -12 z^{3} + 3 z^{2} \) and find the GCF. The coefficient of the first term is -12 and the power of \(z\) is 3. The coefficient of the second term is 3 and the power of \(z\) is 2. The GCF is therefore \(3z^{2}\), as that is what both terms can be divided by completely.

Step 2: Divide the Expression by the GCF

Divide the entire expression by the identified GCF, i.e., \(3z^{2}\), hence \( -12 z^{3}/(3 z^{2}) + 3 z^{2}/(3 z^{2}) \), which simplifies to \( -4 z + 1 \).

Step 3: Combine the GCF and Simplified Expression

Combine the GCF from step 1 with the simplified expression from step 2 to obtain the fully factored expression. The factored result is therefore \(3z^{2}( -4 z + 1 )\).

Key Concepts

Understanding the Greatest Common Factor (GCF)

The concept of the Greatest Common Factor (GCF) is essential in simplifying algebraic expressions. The GCF refers to the largest factor that divides all terms in a polynomial without leaving any remainder. This concept is crucial when factoring expressions, as it helps simplify the expression into a more manageable form.

When identifying the GCF, you need to look at both the coefficients (numbers in front of the variables) and the variables themselves in each term. Here's how you can find the GCF:

• Identify the smallest power of each common variable across all terms. For instance, in the expression \(-12z^3 + 3z^2\), both terms have a 'z' variable, so you choose the lower power, which is \(z^2\).
• Determine the greatest common divisor (GCD) of the numerical coefficients. For \(-12\) and \(3\), the GCD is \(3\).

Combining the GCD of the coefficients with the smallest common power of the variables gives you the complete GCF, which in this case is \(3z^2\). This factorization step is essential because it reduces complexity by simplifying the expression.

Exploring Polynomial Expressions

Polynomial expressions are a fundamental part of algebra. They consist of variables and coefficients, combined using operations like addition, subtraction, and multiplication. Understanding polynomial expressions is crucial because they are at the heart of various algebraic operations and solutions.

A polynomial can be simple or complex, depending on the number and degree of terms involved. For instance, the expression \(-12z^3 + 3z^2\) is a polynomial with two terms. Each term includes a coefficient and a variable raised to a power. Here are some key components of polynomial expressions:

• Terms: Each part of a polynomial separated by a plus or minus sign. In our expression, these are \(-12z^3\) and \(3z^2\).
• Degree: The greatest power of the variable in the polynomial. The expression \(-12z^3 + 3z^2\) has a degree of 3 because the highest power of \(z\) is 3.
• Coefficients: Numerical factors attached to the variables. In our example, \(-12\) and \(3\) are the coefficients.

Grasping these components helps in operations like factoring, adding, subtracting, and simplifying polynomials. This knowledge is foundational for advanced mathematical concepts.

Algebraic Manipulation Techniques

Algebraic manipulation involves a set of strategies to transform and simplify expressions. These techniques are foundational for solving equations and factoring, as seen in problems like factoring the polynomial expression \(-12z^3 + 3z^2\).

Key strategies for algebraic manipulation include:

• Factoring: The process of expressing a polynomial as a product of its factors. For example, in our problem, starting with the expression and applying factorization using the GCF helps simplify it.
• Distribution: Often called the distributive property, where you expand expressions like \(a(b + c)\) into \(ab + ac\). This is frequently used in reverse during factoring.
• Simplification: Reducing expressions to their simplest form by combining like terms or using GCF to factor out common terms, leaving a simpler, equivalent expression.

Using these techniques, you begin by identifying parts of the expression that can be stripped down, such as using the GCF to reduce the expression \(-12z^3 + 3z^2\), factor it into \(3z^2(-4z + 1)\), and thus greatly simplify the problem. Learning these techniques opens up the ability to solve a wide range of algebraic problems.