Problem 50 - Elementary and Intermediate Algebra

Question

Factor. See Example 7 or Example \(12 .\) $$4 b^{2}+12 b-16$$

Step-by-Step Solution

Answer

The factored form is \(4(b + 4)(b - 1)\).

Step 1: Identify coefficient patterns

Notice that the given quadratic is in the form \(ax^2 + bx + c\). Here, \(a = 4\), \(b = 12\), and \(c = -16\).

Step 2: Find the greatest common factor (GCF)

Determine the greatest common factor of the terms: \(4b^2\), \(12b\), and \(-16\). All these numbers are divisible by 4, so the GCF is 4. Factor 4 out of the entire expression:\[ 4(b^2 + 3b - 4) \]

Step 3: Factor the quadratic expression inside the parenthesis

We need to factor \(b^2 + 3b - 4\). Look for two numbers that multiply to \(-4\) (the constant term) and add to \(3\) (the coefficient of \(b\)). These numbers are \(4\) and \(-1\).

Step 4: Write the quadratic in factored form

Using these numbers, we write:\[ b^2 + 3b - 4 = (b + 4)(b - 1) \]Now substitute this back into the expression:\[ 4(b + 4)(b - 1) \]

Step 5: Verify the factorization

Expand \( (b + 4)(b - 1) \) to check correctness:\[ b^2 - b + 4b - 4 = b^2 + 3b - 4 \]The expression matches our original quadratic term, confirming the factorization is correct.

Key Concepts

Greatest Common Factor

Understanding the Greatest Common Factor (GCF) is crucial when factoring quadratic expressions. The GCF is the largest positive integer that divides each of the coefficients of the terms in your polynomial without leaving a remainder. In the expression \(4b^2 + 12b - 16\), identifying the GCF helps to simplify the equation before proceeding with the factoring process.
To find the GCF, we first consider each term separately:

• The term \(4b^2\) has a coefficient of 4.
• The term \(12b\) has a coefficient of 12.
• The constant term is \(-16\).

We note that all the coefficients (4, 12, and -16) are divisible by 4. This means 4 is the GCF for this expression. By factoring out 4 from each term, we can rewrite the original expression as \(4(b^2 + 3b - 4)\).
Finding and using the GCF simplifies our work moving forward, making the quadratic expression easier to handle in subsequent steps.

Quadratic Expression

A quadratic expression is a type of polynomial that takes the general form of \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is a variable. In our example, the quadratic expression is \(b^2 + 3b - 4\), which is derived after factoring out the GCF from the original expression.
Factoring quadratic expressions involves transforming this polynomial into a product of linear factors. This step is crucial because it breaks down complex polynomials into simpler, more manageable pieces. Quadratic expressions will often require identifying two numbers that multiply to give the constant term (here, \(-4\)) and add up to the coefficient of the linear term (here, \(3\)).
Once these numbers (4 and -1 for this expression) are found, they are used to split the middle term and re-group the expression, eventually leading to its completely factored state.

Factored Form

Achieving the factored form of a quadratic expression is the endpoint of the factoring process and involves expressing the original equation as a product of simpler terms. In the problem at hand, after dealing with the quadratic \(b^2 + 3b - 4\), we found the factors 4 and -1 that satisfy the multiplication and addition criteria for factoring.
This led us to express the quadratic as \((b + 4)(b - 1)\). When these expressions are multiplied together, they reproduce the interior quadratic term, confirming their correctness.
Including the GCF we factored out initially, the fully factored form of the expression is \(4(b + 4)(b - 1)\). This form is advantageous as it simplifies solving equations, checking roots, and understanding the behavior of the quadratic function. In this meticulously orchestrated process, moving from an original polynomial expression to its factored form provides clarity and simplicity, opening up multiple avenues for further mathematical applications.