Problem 42 - Introductory Algebra for College Students

Question

Use the method of your choice to factor each trinomial, or state that the trinomial is prime. Check each factorization using FOIL multiplication. $$15 x^{2}-19 x+6$$

Step-by-Step Solution

Answer

The factorized form of the trinomial \(15x^2 - 19x + 6\) is \((5x - 3)(3x - 2)\)

Step 1: Finding the right combination of numbers

First, find the two numbers that add to give -19 and multiply to give \(15*6 = 90\). After investigating potential pairings, the numbers -10 and -9 fit these criteria because \(-10 * -9 = 90\) and \(-10 + -9 = -19\).

Step 2: Factorize the trinomial

Next, rewrite the middle term by splitting it into two terms. This can be achieved by breaking -19x into -10x and -9x. This transformation changes the trinomial into four terms: \(15x^2 - 10x - 9x + 6\). Now, factor by grouping, pair the first two terms together and the last two terms together and factor out from each pair: \(5x(3x - 2) - 3(3x - 2)\). As both terms have a common factor of (3x - 2), pull that common factor out to get the final factorized form: \((5x - 3)(3x - 2)\).

Step 3: Check the factorization using FOIL

To confirm that the factorization process is correct, implement the FOIL method: \((5x - 3)(3x - 2) = 15x^2 - 10x - 9x + 6 = 15x^2 - 19x + 6\). The expression matches the original trinomial, so the factorization process was done correctly.

Key Concepts

FOIL Method

The FOIL method is a process used in algebra to multiply two binomials together. The acronym FOIL stands for First, Outer, Inner, Last, which represents the order in which you multiply the terms.

For instance, let's take two binomials, \(a + b\) and \(c + d\). The FOIL method would unfold as follows:

• First: Multiply the first terms of each binomial, \(a*c\)
• Outer: Multiply the outer terms, \(a*d\)
• Inner: Multiply the inner terms, \(b*c\)
• Last: Multiply the last terms of each binomial, \(b*d\)

The results of these multiplications are then added together to get the product: \(ac + ad + bc + bd\).
In the context of our exercise, after factoring the trinomial \(15x^2-19x+6\), the FOIL method is utilized to verify that \(5x - 3\) and \(3x - 2\) are indeed the correct factors. Multiplying \(5x - 3\) by \(3x - 2\) gives us the original trinomial, confirming the factorization.

Factor by Grouping

Factor by grouping is a technique used to factor polynomials with four or more terms into smaller binomial factors. This method involves grouping terms with common factors and then finding a binomial that they share.

Here's how it's done:

• Identify groups of terms that have a common factor.
• Factor out the common factor from each group.
• If done correctly, a common binomial factor should emerge from each group.
• Finally, factor out the common binomial to find the expression's factors.

By applying this technique to the trinomial \(15x^2-19x+6\), we break it into four terms by splitting the middle term, then group them to factor by grouping: \(15x^2 - 10x\) and \( -9x + 6\). Factoring out the greatest common factors from each group, \(5x\) and \( -3\) respectively, reveals the common binomial factor \(3x - 2\), leading to the factorized form of the expression.

Polynomial Factorization

Polynomial factorization is the process of expressing a polynomial as the product of its factors, which are polynomials of lower degrees. The goal is to break down a complex expression into simpler, multiplyable units.

To factor a polynomial:

• Identify the greatest common factor (GCF) of all terms, if there is one, and factor it out.
• Examine the remaining polynomial to determine if it is a special product, such as a difference of squares or a perfect square trinomial.
• If not, apply methods such as factoring by grouping or seeking binomials that multiply to give the polynomial.

Our exercise required the factorization of a trinomial, a specific type of polynomial. The technique used—splitting the middle term and factoring by grouping—allows the polynomial \(15x^2-19x+6\) to be expressed as \(5x-3\) and \(3x-2\), its factorized components.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, and division) that represent a specific value. These expressions can be simple or complex, containing multiple terms and powers.

When working with algebraic expressions such as trinomials, understanding how to manipulate and factor them is crucial. The process of factoring, which includes the FOIL method, factoring by grouping, and polynomial factorization, simplifies expressions and solves equations. For example, factoring the trinomial \(15x^2-19x+6\) involves recognizing patterns and applying strategies to rewrite the complex expression as a product of simpler expressions, enhancing one's ability to work with and understand algebra.