Problem 45 - Elementary and Intermediate Algebra

Question

Factor by grouping. $$ x^{2}+y x-x-y $$

Step-by-Step Solution

Answer

The factored form is \((x+y)(x-1)\).

Step 1: Group Terms

The first step in factoring by grouping is to divide the polynomial into two pairs: \[ x^{2} + yx \] and \[ -x - y \]. Ensure that you keep the signs intact when grouping the terms.

Step 2: Factor Each Group

Now, factor out the greatest common factor from each group:- For the first group \( x^{2} + yx \), factor out an \( x \): \[ x(x+y) \]- For the second group \( -x - y \), factor out a \( -1 \): \[ -1(x+y) \]

Step 3: Combine the Factors

Notice that each group contains a common binomial factor \((x + y)\). Combine these to write the expression as the product of two factors:\[ (x+y)(x-1) \]

Step 4: Verification

Verify your solution by expanding \((x+y)(x-1)\) to ensure it equals the original expression.- Expand: \( (x+y)(x-1) = x(x) + x(-1) + y(x) + y(-1) = x^2 - x + yx - y \)- The expanded expression matches the original, confirming the factorization is correct.

Key Concepts

Grouping Method

The grouping method is a convenient technique to factor polynomials, especially those with four terms. It involves arranging the polynomial into pairs of terms, which can separately be factored. By grouping carefully, it allows for a common factor to be revealed, simplifying the polynomial into a product of smaller expressions.

To use the grouping method:

• Identify natural pairs of terms in the polynomial.
• Ensure the signs are properly maintained in each term.
• Factor out any common factors from each pair.
• Look for a common binomial factor that can be extracted.

By using the grouping method effectively, you can break complex polynomials into simpler, solvable expressions, making them much easier to handle.

Greatest Common Factor

The Greatest Common Factor (GCF) is a crucial component in polynomial factoring. It is the largest expression that can evenly divide all terms within a polynomial. Identifying and factoring out the GCF simplifies the polynomial and is an essential step in many factoring methods.

Steps to determine the GCF in a polynomial:

• List all terms of the polynomial.
• Determine the factors of each term.
• Identify the largest factor common to all terms.

For example, in the first group of the exercise, the GCF is `x`. Factoring out the GCF simplifies the expression which reveals common factors between the pairs, aiding in further simplification.

Binomial Factor

A binomial factor is a two-term expression that frequently emerges during factoring and is pivotal when using the grouping method. Identifying a common binomial factor allows for efficient simplification of polynomials.

In our exercise, after factoring the groups, we identified `(x + y)` as a common binomial factor. Here's how:

• Factor each group separately.
• Observe any binomial form that appears identical in each group.

The consistency of the binomial factor across the groups suggests that it can be factored out of the entire expression, leading to further simplification.

Polynomial Verification

Verification is the final step in polynomial factoring and it confirms the correctness of the factorization process. By expanding the factored polynomial back into original terms, you ensure that the factorization holds true.

Consider the expression obtained after factoring: `(x+y)(x-1)`. Verification involves:

• Expanding `(x+y)(x-1)` using the distributive property:
• Calculate: `x(x)`, `x(-1)`, `y(x)`, and `y(-1)`.
• Sum them to get: `x^2 - x + yx - y`.

The expanded polynomial should match the original polynomial. This final check ensures no errors were made in the simplification process, thereby validating the solution.