Problem 26 - A Graphical Approach to College Algebra

Question

Factor each trinomial completely. $$5 a^{2}-7 a b-6 b^{2}$$

Step-by-Step Solution

Answer

The trinomial factors to \((5a + 3b)(a - 2b)\).

Step 1: Identify a, b, and c in the Quadratic Trinomial

The given trinomial is in the form \( ax^2 + bx + c \), where \( a = 5 \), \( b = -7 \), and \( c = -6 \). In terms of variable, it is \( 5a^2 - 7ab - 6b^2 \).

Step 2: Multiply a and c

Multiply the coefficient of the squared term (5) with the constant term (-6), giving you \( 5 \times (-6) = -30 \).

Step 3: Find Two Numbers that Multiply to ac and Add to b

We need to find two numbers that multiply to -30 and add to -7. These numbers are -10 and +3 because \( -10 \times 3 = -30 \) and \( -10 + 3 = -7 \).

Step 4: Rewrite Middle Term Using the Two Numbers

Rewrite the trinomial by splitting the middle term using the two numbers found: \( 5a^2 - 10ab + 3ab - 6b^2 \).

Step 5: Factor by Grouping

Group the terms: \( (5a^2 - 10ab) + (3ab - 6b^2) \).

Step 6: Factor Out the Greatest Common Factor from Each Group

In the first group \( (5a^2 - 10ab) \), the greatest common factor is \( 5a \), so factor it out: \( 5a(a - 2b) \). In the second group \( (3ab - 6b^2) \), the greatest common factor is \( 3b \), so factor it out: \( 3b(a - 2b) \).

Step 7: Combine the Factored Groups

Now that both groups have \( (a - 2b) \), factor this common term out: \((5a + 3b)(a - 2b)\). The completely factored form is \((5a + 3b)(a - 2b)\).

Key Concepts

Quadratic Trinomial

A quadratic trinomial is an algebraic expression consisting of three terms. It has the general form \( ax^2 + bx + c \), where:

• \( a \), \( b \), and \( c \) are constants, usually real numbers.
• \( x \) is the variable, and \( x^2 \) is its squared term.

In our example, the quadratic trinomial is \( 5a^2 - 7ab - 6b^2 \). The term \( 5a^2 \) is the quadratic term, \(-7ab\) is the linear term, and \(-6b^2\) is the constant term expressed in terms of \( b^2 \). The purpose of factoring a quadratic trinomial is to express it as a product of two binomials, simplifying further calculations or solving equations.

Greatest Common Factor

The concept of the greatest common factor (GCF) involves finding the largest factor that is common within two or more numbers or expressions. For factoring purposes, it helps to simplify expressions by pulling out this common factor.
In our exercise, we focus on factoring out the GCF from different grouped terms. When examining the expressions \( (5a^2 - 10ab) \) and \( (3ab - 6b^2) \):

• For \( 5a^2 - 10ab \), the GCF is \( 5a \). By factoring this out, we get \( 5a(a - 2b) \).
• For \( 3ab - 6b^2 \), the GCF is \( 3b \). By factoring this out, we derive \( 3b(a - 2b) \).

Identifying and pulling out the GCF simplifies grouping terms and facilitates further factoring.

Factor by Grouping

Factor by grouping is a strategy used in factoring polynomials, which means reorganizing terms to reveal common factors you can utilize. It’s particularly handy in factorizations of quadratic trinomials or when the polynomial has four terms.
In the step-by-step breakdown used for our initial trinomial example, we:

• Rewrote and split terms to create two groups: \( (5a^2 - 10ab) \) and \( (3ab - 6b^2) \).
• Found and factored out the GCF from each group. The term \( a - 2b \) appeared in both factored groups.

With both groups having the common binomial \( a - 2b \), it was factored out, resulting in \((5a + 3b)(a - 2b)\). Factoring by grouping provides a clear path by simplifying the expression into solvable or evaluable binomials.