Problem 42 - Elementary and Intermediate Algebra

Question

Factor. If a polynomial can't be factored, write "prime." $$ 49-w^{2} $$

Step-by-Step Solution

Answer

The polynomial 49 - w^2 factors to (7-w)(7+w).

Step 1: Recognize the Difference of Squares

The given expression is \(49 - w^2\). Notice that 49 is a perfect square since \(49 = 7^2\) and \(w^2\) is also a perfect square because it is \((w)^2\). The expression can be rewritten as \(a^2 - b^2\), where \(a = 7\) and \(b = w\).

Step 2: Apply the Difference of Squares Formula

The difference of squares can be factored using the formula: \(a^2 - b^2 = (a-b)(a+b)\). Here, substitute \(a = 7\) and \(b = w\) into the formula to factor the expression. So, \(49 - w^2 = (7-w)(7+w)\).

Step 3: Verify the Factorization

To ensure the factorization is correct, expand \((7-w)(7+w)\) and verify that it results in \(49 - w^2\). Expanding we get, \(7 imes 7 + 7 imes w - w imes 7 - w imes w = 49 - w^2\). The factorization is correct as it matches the original polynomial.

Key Concepts

Difference of Squares

The concept of the 'difference of squares' is a useful tool in polynomial factorization. Essentially, it applies when you have two perfect squares separated by a minus sign. This can be written in the mathematical form as \(a^2 - b^2\). Determining whether an expression is a difference of squares is the first step. It should involve

• A square of one term, \(a^2\).
• A square of another term, \(b^2\).
• These squares being subtracted.

For example, in the expression \(49 - w^2\), 49 is \(7^2\), and \(w^2\) is a square by itself. Notice the subtraction in between them, confirming that it is indeed a difference of squares. This understanding forms the foundation for further breaking down complex polynomials effortlessly.When you recognize such an expression, you can rewrite it using the difference of squares formula: \[a^2 - b^2 = (a - b)(a + b)\].This rule is straightforward to apply once the expression is identified correctly. Knowing this can simplify your work significantly in algebraic manipulations.

Perfect Squares

Perfect squares are numbers or expressions derived from squaring a number or variable. Recognizing whether a number or variable expression is a perfect square is key to identifying patterns like the difference of squares. A perfect square takes the form \(x^2\) where x is any integer or algebraic expression. Here are some examples:

• \(4\) is a perfect square because \(4 = 2^2\).
• \(9x^2\) is a perfect square because it can be expressed as \((3x)^2\).
• \((a + b)^2\) signifies a more complex algebraic perfect square.

In the provided exercise, both terms \(49\) and \(w^2\) are perfect squares, represented as \(7^2\) and \(w^2\), respectively. This realization helps demystify the factorization process and allows one to use specific algebraic identities designed for such expressions.By mastering how to identify perfect squares, students can quickly set up their factorization using standardized methods which streamline solving broader polynomial expressions.

Polynomial Factorization

Polynomial factorization is a technique used to express a polynomial as a product of its factors. It involves breaking down complex expressions into simpler, more manipulable parts, which can make equations easier to solve. This process becomes efficient by applying known formulas and recognizing patterns:

• Look for common factors to simplify your polynomial.
• Identify special patterns like difference of squares or trinomial squares.
• Utilize algebraic identities to assist in the factorization where possible.

For instance, taking the polynomial \(49 - w^2\), it was factorized using the difference of squares formula. The formula \(a^2 - b^2 = (a-b)(a+b)\) directly applies, giving the factorization as \((7-w)(7+w)\).Verifying factorization is a vital step in ensuring your solution’s correctness. Simply expand the factored form to check if it matches the original polynomial. This verification shows each step's accuracy and bolsters understanding and confidence in handling homogenous expressions.Grasping the principles behind polynomial factorization can unlock solutions to many mathematical challenges, making it an indispensable skill in a student's algebra toolkit.