Problem 24 - Elementary Algebra

Question

Factor. $$ 12 a_{2}-4 a b-5 b_{2} $$

Step-by-Step Solution

Answer

The expression factors into \((3a + b)(4a - 5b)\).

Step 1: Identify potential factors

The expression given is a trinomial: \(12a^2 - 4ab - 5b^2\). Check for two binomials that can be multiplied to get this expression. We are looking for factors in the form \((ma + nb)(pa + qb)\). Identify the coefficient relationships: \(m \times p = 12\), \(n \times q = -5\), and \(m \times q + n \times p = -4\).

Step 2: Find combinations for the factors

List pairs of factors that multiply to 12 and -5. For 12, we have the pairs: (1, 12), (2, 6), (3, 4). For -5, the pairs can be (-1, 5) or (1, -5). We need the combination that satisfies the middle term \(-4\), which will arise from the sum \(mq + np = -4\).

Step 3: Trial and error method

Try out different combinations from Step 2. Arrive at the use of factors \((3a + 1b)(4a - 5b)\) because when we expand, \((3 \times 4)a^2 + (3 \times -5)ab + (1 \times 4)ab + (1 \times -5)b^2 = 12a^2 - 4ab - 5b^2\).

Step 4: Verify the factoring

Distribute the binomials: \[(3a + 1b)(4a - 5b) = 3a \cdot 4a + 3a \cdot (-5b) + 1b \cdot 4a + 1b \cdot (-5b)\]Simplify:\[= 12a^2 - 15ab + 4ab - 5b^2 = 12a^2 - 4ab - 5b^2\]The expression is correctly factored.

Key Concepts

Trinomial

In algebra, a trinomial is an expression consisting of three terms. These terms are separated by plus or minus signs. For the expression given, that is, \(12a^2 - 4ab - 5b^2\), you can see there are three separate components:

• \(12a^2\)
• \(-4ab\)
• \(-5b^2\)

The purpose of factoring a trinomial is to express it as a product of simpler expressions called binomials. In many cases, factoring allows you to understand characteristics of the expression like its roots or zeros.
This is particularly useful in solving equations or finding graph intercepts.

Binomials

Binomials are algebraic expressions that contain exactly two terms. For example, in the factoring process, we look to express a trinomial as a product of two binomials. This is what happens in the final factorized form: \((3a + 1b)(4a - 5b)\). Both of these terms, \((3a + 1b)\) and \((4a - 5b)\), are binomials.

• "3a + 1b" has two terms: \(3a\) and \(1b\).
• "4a - 5b" has two terms: \(4a\) and \(-5b\).

Factoring trinomials into binomials is a common practice in algebra, particularly for quadratic expressions. It allows the simplified algebraic work, making solving inequalities and equations much easier.
Additionally, understanding the structure of binomials helps in using algebraic identities.

Distributive Property

The distributive property is a fundamental principle in algebra. It lets you multiply a single term by each term within a parentheses group. This property is pivotal in the factorization process to ensure the results are accurate.If you have two binomials, like \((3a + 1b)(4a - 5b)\), the distributive property helps you expand it:

• Multiply **3a** by **4a**: yields \(12a^2\)
• Multiply **3a** by \(-5b\): results in \(-15ab\)
• Multiply **1b** by **4a**: gives \(4ab\)
• Multiply **1b** by \(-5b\): produces \(-5b^2\)

Adding these up, \(12a^2 - 15ab + 4ab - 5b^2\) simplifies back to the original trinomial, confirming the factorization is accurate.
Understanding this property helps in simplifying algebraic expressions and solving equations efficiently.