Problem 17 - College Algebra with Modeling and Visualization

Question

Factor out the greatest common factor:. \(-4 a^{2}-2 a b+6 a b^{2}\)

Step-by-Step Solution

Answer

The factored expression is \(2a(-2a - b + 3b^2)\).

Step 1: Identify Each Term's Coefficients and Variables

First, recognize each term in the expression. The expression is \(-4a^2 - 2ab + 6ab^2\). The coefficient for each term is \(-4\), \(-2\), and \(6\) respectively. The variables involved are \(a\) and \(b\).

Step 2: Determine the Greatest Common Factor of the Coefficients

Next, find the greatest common factor (GCF) of the coefficients \(-4\), \(-2\), and \(6\). The GCF is \(2\) since it's the largest number that divides all three coefficients.

Step 3: Identify the Lowest Power of Each Variable Present

Now, identify the smallest exponent for each variable. For \(a\), both \(-4a^2\) and \(-2ab\) have \(a\), and \(2a^1\) is the smallest power. For \(b\), it's \(0\) since \(-4a^2\) has no \(b\). The GCF for the variables is thus \(a\).

Step 4: Factor out the Greatest Common Factor from the Expression

Combine the GCF of the coefficients (\(2\)) and the variables (\(a\)) to get \(2a\). Divide each term by \(2a\): \(-4a^2 \div 2a = -2a\), \(-2ab \div 2a = -b\), and \(6ab^2 \div 2a = 3b^2\).

Step 5: Write the Factored Expression

Combine the factored terms: the expression becomes \(2a(-2a - b + 3b^2)\). The greatest common factor \(2a\) is factored out, leaving the expression inside the parentheses.

Key Concepts

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest number that can evenly divide a set of numbers without leaving any remainder.
In algebra, it also involves finding the lowest power of the common variables across terms.
For example, let's consider the expression \(-4a^2 - 2ab + 6ab^2\). Each term has coefficients as \(-4\), \(-2\), and \(6\).
To find the GCF of the coefficients, determine the largest integer that divides all coefficients evenly. Here, the GCF of \(-4\), \(-2\), and \(6\) is \(2\), because it is the largest number that divides all of them without remainder.
This concept extends to variables as well by finding the lowest exponent of any variable present in the terms. For example, with variables \(a\) and \(b\), the expression has \(a\) in the smallest power of \(a^1\) (appearing in both \(-4a^2\) and \(-2ab\)). \(b\)'s lowest power is 0 since it is not present in \(-4a^2\).
Therefore, the GCF is \(2a\).

Polynomial Expressions

Polynomial expressions are sums of terms made up from constants and variables raised to non-negative integer powers. Every term in a polynomial can have a coefficient, which is a number multiplying the variables.
The general form of a polynomial expression is represented as \(ax^n + bx^{n-1} + ext{...} + zx^0\), where \(a, b, ext{...}, z\) are coefficients and \(n\) is a non-negative integer.
In the given exercise, the polynomial \(-4a^2 - 2ab + 6ab^2\) consists of three terms:

• The first term is \(-4a^2\), with \(-4\) as a coefficient and \(a\) as the variable raised to the power of 2.
• The second term \(-2ab\) has \(-2\) as a coefficient and variables \(a\) and \(b\), each with their exponents.
• The third term \(6ab^2\) includes \(6\) as its coefficient, with \(a\) raised to 1 and \(b\) raised to 2.

Understanding each term's structure allows you to manipulate and simplify these expressions more effectively.

Factoring Techniques

Factoring techniques in algebra are methods used to rewrite expressions as the product of simpler factors. This process often involves finding and extracting the Greatest Common Factor (GCF), applying special formulas, or recognizing patterns.
In our expression \(-4a^2 - 2ab + 6ab^2\), we start by identifying the GCF, which is \(2a\). Factoring out \(2a\) gives each term divided by \(2a\):

• From \(-4a^2\), dividing by \(2a\) results in \(-2a\).
• For \(-2ab\), dividing by \(2a\) results in \(-b\).
• Lastly, \(6ab^2\) divided by \(2a\) results in \(3b^2\).

Hence, the expression becomes \(2a(-2a - b + 3b^2)\).
Recognizing common factors is a fundamental technique in simplifying polynomial expressions, aiding in solving equations and understanding algebraic functions.