Problem 83 - Introductory Algebra for College Students

Question

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$x y-7 x+3 y-21$$

Step-by-Step Solution

Answer

The factorized form of the polynomial \(x y - 7x + 3y - 21\) is \((x + 3)(y - 7)\).

Step 1: Understanding the polynomial

The given polynomial is \(x y-7 x+3 y-21\). This is a four-term polynomial, and we will try to group them into a set of two-terms each.

Step 2: Grouping terms

Group the terms such to uncover their common factors. In this case we can group the first and second term together as \(xy - 7x\) and the third and fourth term together as \(3y - 21\). So, the polynomial can be rewritten like this: \((xy - 7x) + (3y - 21)\).

Step 3: Factoring expressions

Factor out the common factors in each of these two expressions. For the first expression, \(xy - 7x\), the common factor is \(x\). Factoring out \(x\) , we get: \(x(y - 7)\). For the second expression, \(3y - 21\), the common factor is 3. Factoring out 3, we get: \(3(y - 7)\).

Step 4: Final Factorization

Upon inspection, we notice that \((y - 7)\) is a term common to both expressions after factoring. Therefore, we can further factor out \((y - 7)\) from the equation yielding the final factorized form of the given polynomial. Hence, \(x y - 7x + 3y - 21 = (x + 3)(y - 7)\).

Key Concepts

Polynomials in Several Variables

Polynomials can have more than one kind of variable. When dealing with **polynomials in several variables**, we examine expressions like the one in our example:

• x
• y

These are the three crucial components in our polynomial, which is:

• xy
• - 7x
• + 3y
• - 21

This polynomial showcases terms like xy and combinations of x and y with constants.
This kind of polynomial is common in algebra and requires careful handling since it involves multiple variable interactions.
Understanding how these variables are interconnected is a basic step in factoring them successfully.

Factoring Expressions

**Factoring expressions** refers to breaking down a complex expression into simpler parts that, when multiplied together, give back the original expression.
In our case, we take the expression

• xy - 7x + 3y - 21
• and transform it into something easier to handle: (x + 3)(y - 7).

Factoring is useful for simplification, solving equations, and understanding polynomial properties.
Each factored part of our polynomial is significant because it reveals the essential components that construct the expression.
Think of it as breaking down a recipe into its ingredients to understand better what each contributes to the dish.

Grouping Terms

The **grouping of terms** is a strategic step in the factoring process. To simplify the polynomial, we split it into manageable groups such as

• (xy - 7x)
• (3y - 21),

separated logically based on common factors.
This separation helps in easily uncovering factors common to each group, making the larger expression simpler to manipulate.

• By observing the grouped terms, we notice how they can share a common component, such as (y - 7) in our exercise, which becomes more apparent once terms are grouped effectively.

This clever grouping technique is a helpful trick in the toolbox of polynomial factorization.

Common Factors

Finding **common factors** is like identifying the common thread weaving through different pieces of a larger puzzle.
For example, in the polynomials (xy - 7x) and (3y - 21),

• x is the common factor in the first group
• 3 in the second group.

However, both expressions also share (y - 7) as a factor.
Once these common factors are identified, they can be factored out, simplifying the polynomial significantly.
Recognizing and factoring out common factors is a crucial step in polynomial factorization, facilitating a deeper understanding and easier manipulation of the expressions.
This step is essential because it enables the reduction of the polynomial into simpler, easily solvable parts.