Problem 23 - Technical Mathematics with Calculus

Question

Factor completely, by hand or by calculator. Check your results. The General Quadratic Trinomial. $$12 b^{2}-b-6$$

Step-by-Step Solution

Answer

\((3b + 2)(4b - 3).\)

Step 1: Write Down the Quadratic Trinomial

Begin by writing the given quadratic trinomial: \[12b^2 - b - 6.\]

Step 2: Multiply the Coefficient of the Quadratic Term by the Constant Term

Multiply the coefficient of the quadratic term (12) by the constant term (-6), to find the product: \[12 \times (-6) = -72.\] This product will be used to find two numbers that add up to the coefficient of the linear term (-1).

Step 3: Find Two Numbers That Multiply to -72 and Add to -1

Find two numbers that multiply together to give -72 and add together to give -1. In this case, the numbers are 8 and -9; \[8 \times (-9) = -72 \quad \text{and} \quad 8 + (-9) = -1.\]

Step 4: Rewrite the Linear Term Using the Found Numbers

Rewrite the original quadratic trinomial by breaking up the linear term using the two numbers found in the previous step: \[12b^2 + 8b - 9b - 6.\] This is also known as the method of 'splitting the middle term'.

Step 5: Group the Terms into Two Pairs

Group the terms to form two pairs: \[(12b^2 + 8b) + (-9b - 6).\] Now, we will factor each pair separately.

Step 6: Factor Out the Greatest Common Factor (GCF) of Each Pair

From the first pair, factor out the GCF (4b): \[4b(3b + 2).\]From the second pair, factor out the GCF (-3): \[-3(3b + 2).\] This creates a common binomial factor.

Step 7: Factor Out the Common Binomial Factor

Factor out the common binomial factor of the two groups: \[(3b + 2).\]Now write down the final factored form which includes the factored out terms from the previous step: \[4b(3b + 2) - 3(3b + 2) = (3b + 2)(4b - 3).\]

Step 8: Write Down the Completely Factored Form

Combine the factored out parts from step 6 and the common binomial factor to write the completely factored form of the quadratic trinomial: \[(3b + 2)(4b - 3).\]

Step 9: Check the Factored Form

To check that the factoring is correct, use the distributive property (foil) to expand the factors back into the original trinomial: \[(3b + 2)(4b - 3) = 12b^2 + 8b - 9b - 6 = 12b^2 - b - 6,\] which matches the original trinomial, confirming that the factoring is correct.

Key Concepts

Quadratic Trinomials

Quadratic trinomials are polynomial expressions that contain three terms and are of the form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, with \( a \eq 0\). They describe a parabola when plotted on a graph. Factoring these trinomials into a product of binomials is a critical skill in algebra that simplifies expressions and solves quadratic equations.

The fundamental approach involves looking for two numbers that multiply to the product \( ac \) and add to \( b \)—your linear coefficient. Once these two numbers are found, they are used to reframe the middle term, leading to easier factoring through grouping. The process combines analytical thinking and trial-and-error, and it's always followed by a crucial check - multiplying your factors out to ensure you return to the original quadratic trinomial.

Greatest Common Factor (GCF)

The Greatest Common Factor (GCF) is the highest number that divides exactly into two or more numbers. When factoring expressions, especially polynomial ones, finding the GCF of terms is a fundamental step.

Identifying the GCF enables us to factor it out, leaving behind a simpler expression. It's like simplifying a fraction to its lowest term. In some cases, the GCF may include variables as well as numbers. Factoring out the GCF is especially useful in the method of grouping, where we look at each group of terms to factor out their common factors. This first step can sometimes make or break your attempt to simplify a complex expression, and thus, enhances the overall function simplification or the solution of polynomial equations.

Splitting the Middle Term

Splitting the middle term is one among the methods used in factoring quadratic trinomials. Here, the objective is to decompose the middle term of the trinomial into two terms whose coefficients will make it possible to group and factor by grouping.

Finding the two magical numbers that multiply to \( ac \) and add up to \( b \) might feel like a puzzle, but once the correct pair is discovered, it divides the problem into manageable parts. This step is crucial, as it prepares the equation for the subsequent technique known as factoring by grouping. The key idea is to recognize that not just any two numbers will work - they must be carefully selected to satisfy both multiplying and adding conditions.

Factoring by Grouping

Factoring by grouping is a technique used when no simple factorization is apparent. After splitting the middle term, we can arrange terms into two pairs, each grouping having a common factor that can be factored out.

Upon factoring out the GCF from each pair, a common binomial factor often emerges, which we then also factor out. Grouping doesn't always work—it's contingent on selecting the proper numbers when splitting the middle term.

Application in Our Example

In our problem, after splitting, we grouped \( 12b^2 + 8b \) and \( -9b - 6 \) to factor out \( 4b \) and \( -3 \) respectively, leaving us with the common factor of \( 3b + 2 \). Successfully pairing and factoring out the common binomials simplifies the expression into a product of two factors, achieving the final factored form.