Problem 30 - A Graphical Approach to College Algebra

Question

Factor each trinomial completely. $$24 a^{4}+10 a^{3} b-4 a^{2} b^{2}$$

Step-by-Step Solution

Answer

The complete factorization is \(2a^2(4a - b)(3a + 2b)\).

Step 1: Identify Common Factors

Look at the coefficients of each term in the trinomial and identify if there is a common factor. The terms are \(24a^4, 10a^3b,\) and \(-4a^2b^2\). Notice that each term contains the factor \(2a^2\). So, extract \(2a^2\) from each term:

Step 2: Factor Out the Common Factor

Divide each term by \(2a^2\):\[24a^4 \div 2a^2 = 12a^2\]\[10a^3b \div 2a^2 = 5ab\]\[-4a^2b^2 \div 2a^2 = -2b^2\]This gives us:\[2a^2(12a^2 + 5ab - 2b^2)\]

Step 3: Factor the Quadratic Trinomial

Now, focus on factoring \((12a^2 + 5ab - 2b^2)\). Use the method of grouping or trial and error to split the middle term and factor completely. Try finding two numbers that multiply to \((12a^2) \cdot (-2b^2) = -24a^2b^2\) and add to \(5ab\). These numbers are \(+8ab\) and \(-3ab\).

Step 4: Apply Grouping Method

Rewrite \(12a^2 + 5ab - 2b^2\) as \(12a^2 + 8ab - 3ab - 2b^2\) and group the terms:\[(12a^2 + 8ab) + (-3ab - 2b^2)\]

Step 5: Factor Each Group Separately

For \((12a^2 + 8ab)\), factor out \(4a\):\[4a(3a + 2b)\]For \((-3ab - 2b^2)\), factor out \(-b\):\[-b(3a + 2b)\]We have:\[4a(3a + 2b) - b(3a + 2b)\]

Step 6: Factor Out the Common Expression (

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Notice that \((3a + 2b)\) is a common factor, so factor it out:\[(4a - b)(3a + 2b)\]The complete factorization of the trinomial is:\[2a^2(4a - b)(3a + 2b)\]

Key Concepts

Common Factors

To start factoring trinomials, it's crucial to identify any common factors shared by all terms. In a trinomial like the one given, check each term to find what mathematical components they share. These could be numerical values, variables, or their combinations. For example, given the trinomial \(24a^4 + 10a^3b - 4a^2b^2\):

• Look for the largest common numerical factor. Here, each coefficient pairs with a factor of 2.
• Next, examine the variables and their powers. Each term includes \(a^2\).

Extracting the common factor \(2a^2\) simplifies the trinomial by a division process, leading to a new expression within parentheses. This streamlined trinomial is now easier to work with for further factoring. Paying close attention to common factors is an essential first step in simplifying and eventually completely factoring any trinomial.

Quadratic Trinomial

A quadratic trinomial refers to a three-term algebraic expression with a squared term as its highest degree. The structure often appears as \(ax^2 + bx + c\). In the exercise, after factoring out the common factor \(2a^2\), the new quadratic trinomial to manage is \((12a^2 + 5ab - 2b^2)\).

• It features a leading term \(12a^2\), a middle term \(5ab\), and a constant term \(-2b^2\).
• Identifying a quadratic can signal the potential use of various factoring methods, such as grouping or another technique suitable to the specific structure.

Recognizing a quadratic trinomial is pivotal as it dictates the methods you might utilize next. The goal is to factor the quadratic fully to break it down into simpler linear binomials.

Grouping Method

The grouping method is a strategic approach to factor more complex trinomials. It involves creating groups within the expression to reveal common factors. With our trinomial \((12a^2 + 5ab - 2b^2)\), the middle term \(5ab\) is split into two parts: \(8ab\) and \(-3ab\). This enables grouping:

• Group \((12a^2 + 8ab)\) and \((-3ab - 2b^2)\).
• Within each group, factor out the greatest common factor. For example, \(4a\) from the first group and \(-b\) from the second group.

Once each group is factored, identify any common binomial factor. Here, \((3a + 2b)\) surfaces as a common factor from both groups. Pulling it out yields \((4a - b)(3a + 2b)\), indicating the trinomial is factored fully. The grouping method is powerful for untangling complex expressions, turning a challenging trinomial into manageable components.