Problem 43 - Algebra 2

Question

Factor completely. If the polynomial is not factorable, write prime. $$ y^{4}-z^{2} $$

Step-by-Step Solution

Answer

The polynomial factors to \((y^2 - z)(y^2 + z)\).

Step 1: Recognize the Difference of Squares

The polynomial \( y^4 - z^2 \) resembles the difference of squares pattern \( a^2 - b^2 = (a - b)(a + b) \). Identify \( a^2 = y^4 \) and \( b^2 = z^2 \). This gives us \( a = y^2 \) and \( b = z \).

Step 2: Apply the Difference of Squares Formula

Substitute \( a = y^2 \) and \( b = z \) into the formula: \( y^4 - z^2 = (y^2 - z)(y^2 + z) \).

Step 3: Check for Further Factoring

The polynomial \( (y^2 - z)(y^2 + z) \) consists of terms that cannot be further factored using integers. Thus, it is fully factored at this stage.

Key Concepts

Difference of Squares

The concept of a "difference of squares" is important when factoring polynomials. It stems from the fact that a square number can be subtracted from another square number in a specific way. The formula for this is:

• \(a^2 - b^2 = (a - b)(a + b)\)

Recognizing this pattern can greatly simplify the process of factoring. To identify a difference of squares, check if the expression is a subtraction sign between two perfect squares.
For example, in the expression \(y^4 - z^2\), recognize that \(y^4\) is \((y^2)^2\) and \(z^2\) is \(z^2\), making it a difference of squares. This insight allows quick application of the formula. It's crucial to become familiar with common perfect squares and their roots to speed up this identification.
This particular method only works when both terms are perfect squares and are separated by a subtraction. If they're not perfect squares or if it's an addition, a different factoring strategy might be necessary.

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into a product of simpler polynomials. It's like taking apart a complex Lego structure into smaller, manageable blocks. This helps us solve equations, simplify expressions, and find mathematical roots.

• Always look for a "greatest common factor" to factor out first if possible.
• Recognize special patterns like the difference of squares, perfect square trinomials, or sum/difference of cubes.

In our exercise, we worked specifically with a difference of squares pattern \(y^4 - z^2\). After recognizing the pattern, substitute the terms to use the formula: \((y^2 - z)(y^2 + z)\). This method simplifies the polynomial without altering its value.
Before declaring a polynomial totally factored, check each factor to determine if it can be further simplified. If no further factoring is possible, you're done. Factoring polynomials accurately shows deeper mathematical patterns and relationships that might not be obvious at first glance.

Quadratic Expressions

Quadratic expressions are polynomials that include terms up to the second degree, typically in the form \(ax^2 + bx + c\). However, quadratic-like expressions, even when expressed in higher powers like \(y^4 - z^2\), can often be treated similarly when factoring.
In the exercise, by comparing \(y^4\) with \(a^2 = y^2\), we rewrote the expression as a quadratic \((y^2)^2 - z^2\). This transformation allowed us to use quadratic-like factoring methods, focusing on special patterns.

• Identify and use the structure of quadratic expressions to simplify polynomial operations.
• Even non-traditional quadratics can benefit from quadratic solution techniques, such as factoring by pattern recognition.

When you see expressions with even exponents, think about refactoring them into quadratic forms. This approach simplifies the solution process and helps in solving equations related to motion, area, and several real-world applications.