Problem 73 - Beginning and Intermediate Algebra

Question

Factor by grouping. $$36 g^{4}+3 g h-96 g^{3} h-8 h^{2}$$

Step-by-Step Solution

Answer

The given expression factored by grouping is: \((3g-8h)(12g^3+h) \).

Step 1: Divide the expression into two groups

Divide the given expression into two groups by separating the terms with addition or subtraction: $$ (36g^4 + 3gh) - (96g^3h + 8h^2) $$

Step 2: Factor out the GCF from both groups

Factor out the greatest common factor from the terms within each group: $$ 3g(12g^3 + h) - 8h(12g^3 h + h)$$

Step 3: Determine if a common binomial factor exists

Check if we have a common binomial factor between both expressions. In this case, we have a common binomial factor of \( (12g^3 + h) \): $$ (3g-8h)(12g^3+h) $$ So, the given expression factored by grouping is: $$ (3g-8h)(12g^3+h) $$

Key Concepts

greatest common factor (GCF)

Finding the greatest common factor (GCF) is a fundamental step in polynomial factorization. The GCF is the largest factor that divides all terms of a polynomial without a remainder. When tackling a problem like the one in our exercise, locating the GCF helps to simplify expressions.

• First, look at each group of terms separately.
• Identify any numerical coefficients and variable parts common in the terms within a group.
• The GCF is the highest value or expression that can evenly divide each term of the group.

In our exercise, we first identified the GCF of the first group \( (36g^4 + 3gh) \) to be \( 3g \), and for the second group, \( (96g^3h + 8h^2) \), it turned out to be \( 8h \). By factoring these out, we simplify the expression, making it easier to detect common factors between groups.

grouping method

The grouping method is an efficient technique for factoring polynomials, especially when dealing with four-term expressions. This method involves organizing the expression into smaller 'groups' and then factoring individually before combining them through common factors.

• Start by splitting the polynomial into two smaller groups, usually half of the terms each.
• Find the greatest common factor for each group and factor it out.
• Inspect the factored expressions for any common binomial factors.

Our example begins by splitting into \( (36g^4 + 3gh) \) and \( (96g^3h + 8h^2) \), which allows us to factor out common elements. Once we've isolated these, we can address the entire expression more easily, often discovering a shared binomial factor as seen in this exercise.

algebraic expressions

Algebraic expressions are mathematical phrases that include numbers, variables, and operations. Understanding how to work with algebraic expressions is crucial to rearranging and simplifying equations.These expressions can include:

• Constants: Numbers without variables, such as 8.
• Variables: Symbols, like \( g \) or \( h \), that represent unknown values.
• Coefficients: Numbers placed in front of variables, indicating multiplication.

Algebraic expressions like \( 36g^4 + 3gh - 96g^3h - 8h^2 \) can be complex due to multiple terms with different degrees and coefficients. By applying methods like extracting the GCF and grouping, we simplify their structure, facilitating further manipulation and solution.

binomial factor

Once you've split a polynomial into groups and factored out their GCFs, identifying common binomial factors becomes essential. This step can help in further reducing an expression.A binomial factor consists of two terms linked by either addition or subtraction. For example, in the exercise \( (12g^3 + h) \) emerges as a common binomial factor from both groups.Key aspects include:

• Ensure both groups share an identical binomial factor.
• If they do, you can express the entire polynomial as a product of this binomial and their respective leftover factors.

The factored expression \( (3g-8h)(12g^3+h) \) is a result of recognizing \( (12g^3 + h) \) as a recurring pattern, simplifying the polynomial into a more manageable product form.