Problem 94 - Algebra for College Students

Question

Should help you pull together all of the factoring techniques of this chapter. Factor completely each polynomial, and indicate any that are not factorable using integers. $$3 x y+15 x-2 y-10$$

Step-by-Step Solution

Answer

The complete factorization is \((3x - 2)(y + 5)\). The polynomial is fully factored using integers.

Step 1: Grouping Terms

First, we will group the terms into pairs to factor by grouping. The polynomial is given as:\[ 3xy + 15x - 2y - 10 \]Group it in the following way:\[ (3xy + 15x) + (-2y - 10) \]

Step 2: Factoring Each Group

Next, we factor each group separately:1. For \(3xy + 15x\), factor out the greatest common factor, which is \(3x\): \[ 3x(y + 5) \]2. For \(-2y - 10\), factor out the greatest common factor, which is \(-2\): \[ -2(y + 5) \]

Step 3: Factoring by Grouping

Now, notice that both groups have a common binomial factor of \((y + 5)\). We can factor this common binomial out:\[ (3x - 2)(y + 5) \]

Step 4: Verify Completeness

Verify if the expression has been completely factored by checking each factor to ensure no further factoring is possible.Both \(3x - 2\) and \(y + 5\) don't have factors that can be extracted using integers. Therefore, the factorization is complete.

Key Concepts

Factoring by Grouping

Factoring by grouping is a helpful technique for polynomials with four or more terms. It involves rearranging the polynomial's terms into pairs or groups and then finding common factors in each group.
Let's break it down:

• Step 1: Identify pairs or groups of terms. In our given polynomial, we could pair the terms as \((3xy + 15x) + (-2y - 10)\).
• Step 2: Factor out the common factor from each group separately. For example, \(3xy + 15x\) can have \(3x\) factored out, and \(-2y - 10\) can have \(-2\) factored out.
• Step 3: Look for a common binomial factor. This polynomial shows that both groups share \((y + 5)\).
• Step 4: Write the polynomial as a product of the common binomial factor and another binomial: \((3x - 2)(y + 5)\).

This step-by-step method simplifies finding the factors of a polynomial by reorganizing it into manageable pieces.

Greatest Common Factor

Finding the greatest common factor (GCF) of terms in a polynomial is key to simplifying expressions before further factoring, such as when applying the grouping method.
The GCF is the largest factor shared by two or more terms.
Here's how you determine it:

• Identify the common factors of each term, focusing on coefficients and variables.
• For example, in the group \(3xy + 15x\):
  - The coefficients 3 and 15 share a greatest common factor of 3.
  - The variable factor \(x\) is common to both terms.
• Combine these to get \(3x\), the GCF for this group.

Doing the same for the second group \(-2y - 10\), we identify \(-2\) as the GCF. Extracting these GCFs reduces the polynomial to simpler terms where further factoring, such as grouping, can easily be applied.

Polynomials

Polynomials are algebraic expressions made up of terms involving variables and coefficients, linked together by addition or subtraction.
Understanding polynomials is essential for recognizing patterns for factorization.

• Each term in a polynomial is either a constant or a product involving variables raised to a power, like \(3xy\) in our example.
• The degree of a polynomial is determined by the term with the highest sum of exponents. For instance, in \(3xy + 15x - 2y - 10\), the highest degree is 2 from the term \(3xy\).
• Polynomials are categorized by their degree and the number of terms, such as a quadratic (degree 2) or a binomial (2 terms).

Factoring polynomials involves breaking them down into simpler expressions that can be multiplied to get the original polynomial, using techniques like grouping or finding the greatest common factor.

Integer Factorization

Integer factorization in the context of factoring polynomials refers to simplifying polynomial expressions using integer values.
The principle is ensuring that each factor in the polynomial expression is a whole number value.
Here's why it matters:

• When factoring \(3xy + 15x - 2y - 10\), the goal is to express it as a product of polynomials with integer coefficients, reducing to \((3x - 2)(y + 5)\).
• Integer factorization helps in recognizing results that cannot be simplified further with integer values, confirming that your polynomial is fully factored.
• Polynomials that can't be factored any further using integers are called irreducible or prime over integers.

Understanding integer factorization ensures that your polynomial solutions are expressed in the most simplified, accurate form, preparing the ground for further algebraic manipulations.