Problem 3 - A Graphical Approach to College Algebra

Question

Factor the greatest common factor from each polynomial. $$4 k^{2} m^{3}+8 k^{4} m^{3}-12 k^{2} m^{4}$$

Step-by-Step Solution

Answer

The GCF is \(4k^2m^3\), and the factored form is \(4k^2m^3(1 + 2k^2 - 3m)\).

Step 1: Identify the greatest common factor (GCF) of the coefficients

The coefficients of the terms are 4, 8, and -12. Find the GCF for these numbers. The GCF of 4, 8, and -12 is 4.

Step 2: Identify the common variables and their lowest powers

The common variables in all terms are \(k\) and \(m\). The powers of \(k\) are 2, 4, and 2. The smallest power is 2. The powers of \(m\) are 3, 3, and 4. The smallest power is 3. So, the GCF for the variables is \(k^2 m^3\).

Step 3: Combine GCF of numbers and variables

The GCF of the polynomial is the combination of the GCF of coefficients and variables. So, the GCF is \(4k^2m^3\).

Step 4: Factor out the GCF from each term

Divide each term by the GCF \(4k^2m^3\).1. \[\frac{4k^2m^3}{4k^2m^3} = 1\]2. \[\frac{8k^4m^3}{4k^2m^3} = 2k^2\]3. \[\frac{-12k^2m^4}{4k^2m^3} = -3m\]

Step 5: Write the factored form

Combine the results of Step 4 to rewrite the polynomial:\[4k^2m^3(1 + 2k^2 - 3m)\]

Key Concepts

Greatest Common Factor

The greatest common factor (GCF), also known as the greatest common divisor, is the largest factor that divides two or more numbers without leaving a remainder. When dealing with polynomials, determining the GCF is crucial for simplifying expressions through factoring. The process has two main parts: finding the GCF of the coefficients and the variables.

• **Coefficients**: Begin by identifying the numerical parts of the polynomial. In our example, the coefficients are 4, 8, and -12. To find their GCF, list the divisors for each number and choose the largest common divisor. Here, 4 is the largest number evenly dividing all coefficients.
• **Variables**: Next, look at each variable's exponent across the terms. Identify common variables and select the lowest exponent for each. In our problem, the variables are \(k\) and \(m\). The smallest power of \(k\) is 2, and the smallest for \(m\) is 3.

By combining these, the complete GCF for the given polynomial is \(4k^2m^3\). Understanding how to find the GCF simplifies the factoring of polynomials and is a foundational skill in algebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operation symbols. In algebra, a term is a single part of an expression, which may include constants, variables, and their coefficients multipled together. Recognizing the structure of algebraic expressions makes it easier to handle complex equations and execute operations like factoring.

• **Terms**: Each term in the expression includes a coefficient and variable(s) raised to an exponent. For example, in the term \(4k^2m^3\), 4 is the coefficient, \(k^2\) and \(m^3\) are the variables with exponents.
• **Polynomials**: These are specific algebraic expressions composed of multiple terms. The example expression \(4k^2m^3 + 8k^4m^3 - 12k^2m^4\) is a polynomial because it contains three separate terms.

To work efficiently with algebraic expressions, like simplifying or factoring, one needs to identify similar terms and potential common factors—key steps that allow for rewriting the expression in a simpler form.

Polynomial Division

Polynomial division involves dividing a polynomial by another polynomial or a monomial, often to simplify or reveal certain properties of the expressions. This technique is useful in factoring, as seen when removing the greatest common factor (GCF) to simplify the polynomial.

We utilize polynomial division in simplifying our example expression by removing the GCF. The process here is straightforward, as it involves dividing each term of the polynomial by the GCF:

• For \(4k^2m^3\), divide by itself \((4k^2m^3)\) resulting in 1.
• For \(8k^4m^3\), divide by \(4k^2m^3\) resulting in \(2k^2\).
• For \(-12k^2m^4\), dividing by \(4k^2m^3\) results in \(-3m\).

Finally, the polynomial is expressed as \(4k^2m^3(1 + 2k^2 - 3m)\), showing how polynomial division can simplify expressions. By breaking down polynomials using their GCF, it becomes significantly easier to work with and solve algebraic problems.