Problem 40 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$32 y^{2}+4 y-6$$

Step-by-Step Solution

Answer

The completely factored form of \(32y^2 + 4y - 6\) is \(2(4y - 1)(4y + 3)\).

Step 1: Divide All Terms by the Greatest Common Factor

To make the exercise easier, reduce the terms by dividing them by their greatest common factor. Here, the greatest common factor is 2. Therefore, the equation \(32y^2 + 4y - 6\) becomes \(16y^2 + 2y - 3\).

Step 2: Factor the Quadratic Polynomial

A quadratic equation can be factored into the form \((dy - e)(fy - g)\), where \(d, e, f\) and \(g\) are constants. The constants should be selected such that they multiply to equal \(ac\) and add to equal \(b\). In this case, \(a = 16\), \(b = 2\) and \(c = -3\). After careful selection we obtain the equation as \((4y - 1)(4y + 3)\).

Step 3: Check the Factorization

You should always check the answer by multiplying the factored terms and ensure they equate to the original (simplified) polynomial. Here, \((4y - 1) \times (4y + 3)\) indeed results in the simplified polynomial \(16y^2 + 2y - 3\), attesting that the factoring is correct.

Key Concepts

Greatest Common Factor

The concept of the Greatest Common Factor (GCF) is fundamental when working with polynomials. Understanding it simplifies the process of polynomial factorization significantly.

The GCF is the largest number or expression that divides each term of the polynomial without leaving a remainder. In the polynomial expression given, which is \(32y^2 + 4y - 6\), the greatest common factor among the terms is 2.

What this means is that each term in the polynomial can be divided by 2 without any remainder. By factoring out the GCF, we simplify the polynomial, making it smaller and easier to manage.

• Reduces complexity of the polynomial
• Helps simplify other algebraic operations
• Makes it easier to see other factors of the polynomial

In simpler terms, the GCF is a tool to make our algebraic life easier. It's often the first step in factorization to "clean up" the polynomial before diving into more complex factoring techniques.

Quadratic Polynomial

A quadratic polynomial is a polynomial of degree two, typically in the form of \(ax^2 + bx + c\).

In the exercise, the simplified polynomial \(16y^2 + 2y - 3\) is a quadratic polynomial. Quadratic polynomials can appear simple at a first glance, but factoring them requires a bit of strategy.

Here are key characteristics of a quadratic polynomial:

• It involves a term with a variable squared.
• Has three terms: a quadratic term, a linear term, and a constant.
• The degree of the polynomial is 2, because the highest power of the variable is 2.

The key to factoring these efficiently lies in understanding their structure and how the coefficients relate to each other. Being comfortable with the quadratic form makes transitioning to factoring a lot smoother.

Factoring Techniques

Factoring techniques come in handy when you need to break down a polynomial into simpler parts. There are several methods, and choosing the right one depends on the polynomial form.

In this exercise, factoring the quadratic polynomial \(16y^2 + 2y - 3\) involved finding two binomials \((dy + e)(fy + g)\). The aim is to identify values for \(d, e, f,\) and \(g\) such that:

• The product of \(d \times g\) matches \(a \times c\) (from the equation \(ax^2 + bx + c\)).
• Adding gives the middle term \(b\).

For instance, in the example, \(d\) and \(f\) were both 4, and \(e\) and \(g\) were 1 and -3 respectively to satisfy the conditions.

Verifying the factorization by multiplying the factors back together ensures the accuracy of the factorization. Using these techniques not only simplifies the polynomial but also assists in solving equations and functions simplified by these components.