Problem 50 - Elementary Algebra

Question

Factor. $$ x 2 y 2+25 x y-26 $$

Step-by-Step Solution

Answer

The polynomial \( x^2y^2 + 25xy - 26 \) does not appear to factor into simpler integers.

Step 1: Identify the Polynomial

The polynomial given to us is \( x^2y^2 + 25xy - 26 \). We need to factor this polynomial.

Step 2: Look for a Common Factor

Check if there is a common factor in all the terms of \( x^2y^2 + 25xy - 26 \). Since there is no common variable or number in all three terms, we move to the next step without factoring anything out.

Step 3: Attempt Factoring by Grouping

Since there is no common factor, try factoring by grouping. Split the middle term to assist in finding group pairs, but observe that the middle term cannot be split to match terms. Thus, factoring by grouping won't work here.

Step 4: Check for Other Methods

Usually, for such a polynomial, if typical methods such as grouping or special formulae don't work, this polynomial may not factor into simpler polynomials over the integers. Try using the quadratic formula if it were a quadratic in one variable, but here we suspect the polynomial doesn't factor neatly.

Key Concepts

Common Factor

To factor a polynomial, it's crucial first to look for a common factor. A common factor is a term that can be evenly divided out of each term in the polynomial. However, not every polynomial will have a common factor.
In the example given with the polynomial \(x^2y^2 + 25xy - 26\), we check each term:

• \(x^2y^2\) has factors of \(x\) and \(y\),
• \(25xy\) has factors of \(x\) and \(y\) as well,
• \(-26\) is a numerical term with no variables attached.

A common variable or number must be present in every term to factor it out, which is not the case here. While the first two terms share \(x\) and \(y\), they are absent in \(-26\).
Understanding this process helps you decide when to move to the next step of factoring, confirming that no common factor exists leads us to explore alternative methods.

Factoring by Grouping

Factoring by grouping is often a very handy method when dealing with polynomials. This method involves rearranging and grouping terms to create common factors that can be factored out.
The goal is to split the polynomial into groups that can independently be factored.
If we consider our polynomial \(x^2y^2 + 25xy - 26\), it's tempting to try splitting \(25xy\) to facilitate this process.
Unfortunately, for this polynomial:

• The middle term 25xy can't be split into terms that would allow grouping.
• Without a successful grouping split, the polynomial can't be factored by grouping effectively.

This highlights an important concept for learners: not all polynomials can be conveniently factored by grouping. It's a valuable method, but in cases like this, alternative strategies might be required.

Polynomial Expressions

Polynomial expressions are made up of variables and coefficients, connected through addition, subtraction, and multiplication.
You encounter these expressions in the form of multiple terms, such as in \(x^2y^2 + 25xy - 26\). Understanding the structure of polynomials is essential for effective factoring.
The terms of a polynomial expression may not always offer straightforward factoring options:

• Binomials are often factored by finding simple common factors or applying special patterns like difference of squares.
• Trinomials require methods such as grouping, quadratic factoring, or special formulas.

Our polynomial \(x^2y^2 + 25xy - 26\) is a trinomial. It showcases that some trinomials don't factor neatly, as in this case even classic strategies don't apply effectively.
Recognizing when a polynomial can or cannot be factored over the integers is an important skill. It's necessary to explore all potential methods and be prepared for cases where expressions resist factoring completely.