Problem 68 - Beginning and Intermediate Algebra

Question

Factor completely. $$64 x^{2}-112 x+49$$

Step-by-Step Solution

Answer

The completely factored form of the given expression is: \((2x - 1)(32x - 3)\).

Step 1: Identify the common factors for each term

Let's first identify if there are common factors for each term in the expression: - For \(64x^2\), the factors are: \(2\), \(4\), \(8\), \(16\), \(32\), and \(64\). - For \(112x\), the factors are: \(2\), \(4\), \(7\), \(14\), \(28\), and \(56\). - For \(49\), the factors are: \(1\), \(7\), and \(49\). The greatest common factor for all three terms seems to be 1, therefore we will need to continue to the next step.

Step 2: Group the terms and factor by grouping

Now, let's try to factor the expression by grouping. We split the middle term into two terms, so that each of the resulting groups has a common factor. We notice that \(112x\) can be written as \(32x\cdot3\). So, the expression can be rewritten as: \(64x^2 - 32x\cdot3 + 49\) Now, we can group the terms as: \((64x^2 - 32x) + (- 3x + 49)\)

Step 3: Factor the common factor from each group

Now, we factor out the common factor from each group: \(32x(2x - 1) - 3(2x - 1)\)

Step 4: Factor the common binomial factor

Now, we can see a common binomial factor \((2x - 1)\) in both terms. Let's factor that out as well: \((2x - 1)(32x - 3)\) Our quadratic trinomial is now factored completely as: \((2x - 1)(32x - 3)\)

Key Concepts

Quadratic Trinomials

A quadratic trinomial is a polynomial with three terms, typically expressed in the standard form as \(ax^2 + bx + c\), where:

• \(a\), \(b\), and \(c\) are constants
• \(x\) is the variable
• The degree of the polynomial is 2, given by the highest power of \(x\)

In this exercise, the quadratic trinomial is \(64x^2 - 112x + 49\). Identifying the structure of a quadratic trinomial is crucial in the process of factoring polynomials. Recognizing the terms and their coefficients allows us to apply various factoring techniques.
Each term contributes to the overall form and dictates the strategies used to factor it effectively. For instance, identifying that our trinomial fits the quadratic format helps us look for perfect square patterns or suitable grouping techniques to factor completely.

Greatest Common Factor

The Greatest Common Factor (GCF) is the largest factor that divides all coefficients of the terms in the polynomial. Finding the GCF is an essential first step in simplifying expressions. It helps simplify the expression early on, reducing complexity.
By examining the polynomial given, \(64x^2 - 112x + 49\), you check the coefficients of each term:

• 64 for \(64x^2\)
• 112 for \(-112x\)
• 49 for the constant term

We list the factors of each coefficient and find the greatest one they share. However, if the terms don't have a common factor greater than 1, as in this case, the GCF is 1. Therefore, we move to other methods like factoring by grouping.
Even if the GCF is trivial, performing this check can highlight opportunities for simplifying specific terms of the polynomial.

Factor by Grouping

Factoring by grouping is a technique used when the quadratic trinomial can be split into smaller expressions that reveal common factors. This involves rearranging or rewriting the trinomial in such a way that the expression becomes easier to manage.
Here, we attempt to break down \(64x^2 - 112x + 49\) by finding a smarter way to manage the middle term \(-112x\). It is split into terms like \((64x^2 - 32x) + (-3x + 49)\), so it pairs with a partner for easier factoring.

• The first group \(64x^2 - 32x\) reveals a factor of \(32x\)
• The second group \(-3x + 49\) has no visible common factor, but matches a potential shared expression, leading to \((2x - 1)(32x - 3)\)

By focusing on creating pairs, this method exploits common factors within paired terms to extract and simplify, creating an approach feasible for quaratic expressions with specific arrangements.