Problem 52 - Algebra A Combined Function

Question

Factor each trinomial completely. See Examples 1 through 7. \(54 a^{2}+39 a b-8 b^{2}\)

Step-by-Step Solution

Answer

The factored form is \((9a + 8b)(6a - b)\).

Step 1: Identify the form of the trinomial

The given trinomial is in the form of a quadratic expression: \(54a^2 + 39ab - 8b^2\). We need to factor it completely, recognizing it resembles a quadratic expression in terms of \(a\).

Step 2: Use factoring by grouping

We need to split the middle term \(39ab\) in such a way that the trinomial can be grouped into two binomials. To do this, find two numbers whose product is the coefficient of \(a^2\) and \(b^2\) terms multiplied (\(54 \times -8 = -432\)) and whose sum is \(39\) (the coefficient of \(ab\)). These numbers are \(48\) and \(-9\).

Step 3: Rewrite the trinomial with grouped terms

Using the numbers \(48\) and \(-9\), rewrite the trinomial: \[54a^2 + 48ab - 9ab - 8b^2\]. This allows us to factor by grouping the terms.

Step 4: Factor by grouping

Group the terms into two pairs: \((54a^2 + 48ab) + (-9ab - 8b^2)\). Factor out the greatest common factor from each pair: \(6a(9a + 8b) - b(9a + 8b)\).

Step 5: Complete the grouping factorization

Notice \(9a + 8b\) is a common factor. Factor it out: \((9a + 8b)(6a - b)\). This gives us the completely factored form of the trinomial.

Key Concepts

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically having the form \(ax^2 + bx + c\). The expression we started with, \(54a^2 + 39ab - 8b^2\), is a quadratic in terms of \(a\), because of the \(a^2\) term. Quadratics can have various forms or disguises, such as the one in our exercise involving the term \(ab\), which can make them seem more complex. However, recognizing it as a quadratic is crucial.

Key features of quadratic expressions include:

• The presence of a squared term, which establishes it as a second-degree polynomial.
• The possibility of having single, double, or even triple variables, which influences how we factorize or solve them.

Understanding the structure helps in identifying the right method to factorize or solve the expression efficiently.

Factoring by Grouping

Factoring by grouping is a clever technique where we manipulate an expression to create common factors that can be grouped into pairs of terms. In our situation, the quadratic trinomial \(54a^2 + 39ab - 8b^2\) required us to split the middle term \(39ab\) into two terms, which made the expression easier to handle.

The process works as follows:

• First, we need numbers that multiply to give the product of the leading coefficient and the constant term. Here, \(54\times -8 = -432\).
• Then, we find two numbers that add up to the middle coefficient, \(39\). In this case, those numbers are \(48\) and \(-9\).
• Rewrite the middle term using these two numbers, resulting in \(54a^2 + 48ab - 9ab - 8b^2\).
• Group into pairs: \((54a^2 + 48ab) + (-9ab - 8b^2)\) and factor them separately.

This method effectively prepares the expression for further factoring, leveraging the common factors between pairs to simplify the trinomial.

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. When factoring expressions, obtaining the GCF is a crucial initial step to simplify and eventually solve the polynomial effectively. In the context of our exercise, finding the GCF allows one to see how the trinomial can be split conveniently.

To find the GCF in a grouped expression:

• Look at the coefficients and variables independently, identifying the highest number that evenly divides all coefficients within the group.
• Identify the smallest power of the variable present in each term.
• Factor out this GCF from each group. For example, from \(54a^2 + 48ab\), the GCF is \(6a\), leading to the term \(6a(9a + 8b)\).
• For \(-9ab - 8b^2\), the GCF is \(-b\), simplifying it to \(-b(9a + 8b)\).

Identifying and factoring out the GCF is essential in simplifying complex polynomials to reveal their simplest form or to solve the equation.