Problem 29 - Precalculus: Functions and Graphs

Question

Factor the polynomial. $$x^{2}-49 y^{2}-14 x+49$$

Step-by-Step Solution

Answer

The polynomial factors to \((x-7-7y)(x-7+7y)\).

Step 1: Recognize the Difference of Squares

The polynomial begins with a term \(x^2\) and ends with a term \(-49y^2\). These are perfect squares \((x^2)\) and \((7y)^2\) respectively, indicating a difference of squares: \(x^2 - (7y)^2\). To further factor, look at the polynomial as \((x+7y)(x-7y)\).

Step 2: Reformat the Expression

Notice the middle terms \(-14x + 49\) that we need to account for with common factoring techniques. Observe that \(2(7x)\) equals \(-14x\), suggesting we should express the middle as a result of the entire factorization.

Step 3: Group Terms Appropriately

Rearrange and group the polynomial like so: \((x^2 -14x + 49) - 49y^2\). The first part is a trinomial \(x^2 - 14x + 49\) and can be factored further.

Step 4: Factor the Perfect Square Trinomial

The terms \(x^2 - 14x + 49\) factor to \((x-7)^2\) because it is a perfect square trinomial: \((x-7)(x-7)\).

Step 5: Combine with the Difference of Squares

Substitute back into the difference of squares from Step 1. The expression becomes \(((x-7)-7y)((x-7)+7y)\).

Step 6: Simplify and Finalize the Factorization

Further simplify the factored form to \((x-7 - 7y)(x-7 + 7y)\), confirming that it's perfectly factored.

Key Concepts

Difference of Squares

The difference of squares is a specific algebraic identity used often in polynomial factorization. It applies when you have two perfect squares separated by a subtraction sign. The formula is given by:

• If you have two terms like \(a^2 - b^2\),

you can factor them as \((a-b)(a+b)\).
For example, if you have \(x^2 - 49y^2\), you recognize \(x^2\) as \((x)^2\) and \(49y^2\) as \((7y)^2\).
Therefore, it can be rewritten as \((x-7y)(x+7y)\). This identity is very useful as it simplifies what appears to be complex polynomial expressions, making them easier to work with.

Perfect Square Trinomial

A perfect square trinomial is another type of expression that is particularly easy to factor. These occur when a trinomial (a polynomial with three terms) is the square of a binomial.

• For instance,\(a^2 - 2ab + b^2\) can be expressed as \((a-b)^2\).

In the exercise, the terms\(x^2 - 14x + 49\) are perfectly square as they can be factored down to \((x-7)^2\).
This pattern follows the general form,\(a^2 - 2ab + b^2\) = \((a-b)^2\). Here, \(a = x\) and \(b = 7\). Recognizing a perfect square trinomial is handy since it instantly reduces the expression to a squared binomial, streamlining polynomial factorization.

Factoring Techniques

Factoring techniques are essential when simplifying polynomial expressions. These methods apply different strategies based on the structure of the polynomial in question. Here are some common techniques:

• Grouping: Useful for expressions with four terms.
• Factoring by common terms: Extracts common factors from terms.
• Difference of squares: As explained above, used when identifying squares separated by subtraction.
• Perfect square trinomial factorization: Reduces trinomials that follow a specific squared pattern.

Each technique looks for specific patterns or structures that allow for efficient factorization. By recognizing these patterns, polynomial expressions, no matter how complex they seem, can often be broken down into simpler, more manageable forms.

Polynomial Expressions

Polynomial expressions are foundational in algebra, consisting of variables raised to various powers and multiplied by coefficients. They can take various forms, ranging from simple linear expressions to more complex multi-term polynomials. Understanding polynomials involves:

• Recognizing standard forms: Such as monomials, binomials, and trinomials.
• Operations on polynomials: Like addition, subtraction, multiplication, and division.
• Factoring polynomials: Simplifying them into products of simpler polynomials where possible.

In the given exercise, practicing how to factor polynomials using the difference of squares and the recognition of perfect square trinomials illustrates the manipulation of these expressions.
By mastering such techniques, you gain a powerful toolset for solving a broad range of algebraic problems effectively. This understanding forms the basis of more advanced mathematical topics.