Problem 62 - Elementary and Intermediate Algebra

Question

Factor. $$ y^{4}-625 $$

Step-by-Step Solution

Answer

The expression factors to \((y - 5)(y + 5)(y^2 + 25)\).

Step 1: Recognize the difference of squares

The expression given is similar to a difference of squares, on the form of \( a^2 - b^2 \). Here, notice that \( y^4 - 625 \) can be rewritten as \((y^2)^2 - 25^2 \). This helps in factoring the expression.

Step 2: Apply the difference of squares formula

The difference of squares formula states that \( a^2 - b^2 = (a-b)(a+b) \). Apply this formula to \((y^2)^2 - 25^2\) by letting \( a = y^2 \) and \( b = 25 \), so \( y^4 - 625 = (y^2 - 25)(y^2 + 25) \).

Step 3: Further factor the difference of squares

The term \( y^2 - 25 \) is again a difference of squares, \((y)^2 - 5^2\), which can be factored again using the formula. This gives \( (y - 5)(y + 5) \).

Step 4: Combine all the factors

Substitute back the factored terms from the previous step to combine and arrive at the final factored form: \( (y^2 - 25)(y^2 + 25) = (y - 5)(y + 5)(y^2 + 25) \).

Key Concepts

Difference of squares

The concept of the "difference of squares" is vital in polynomial factorization. It applies when we have an expression of the form \(a^2 - b^2\). This expression can be factored using a simple and elegant formula:

• \( a^2 - b^2 = (a-b)(a+b) \)

This formula works because when expanded, the middle terms cancel each other out. To factor using the difference of squares, identify terms that can be squared to match this pattern. In our exercise, \(y^4 - 625\) can be rearranged into \((y^2)^2 - 25^2\), making it suitable for applying the difference of squares formula. Recognizing these patterns simplifies reformulating complex algebraic expressions into manageable pieces.

Algebraic expressions

Algebraic expressions consist of numbers, variables, and operations combined in a meaningful way. In our example, an expression like \(y^4 - 625\) integrates variables and constants. Understanding algebraic expressions enables us to apply concepts like the difference of squares for simplification.

• The variable \(y\) is raised to a power, demonstrating the polynomial's degree, which dictates the complexity of factoring.
• Expressions may also contain constants, such as 625, which represent fixed values.

By recognizing how variables and constants form an expression, we can efficiently deploy appropriate factoring techniques. A solid understanding of these elements transforms challenging polynomial tasks into straightforward procedures.

Polynomial factorization

Polynomial factorization involves breaking down a polynomial into simpler factors, specifically products of polynomials. It helps easily solve polynomial equations by identifying the roots or zeros. In the exercise \(y^4 - 625\), we demonstrate this by breaking it into factors.

• First, we recognize the difference of squares to factor \((y^2)^2 - 25^2\) into \((y^2 - 25)(y^2 + 25)\).
• Next, further decompose the factor \(y^2 - 25\) into \((y - 5)(y + 5)\), again using the difference of squares pattern.

The final factored form \((y - 5)(y + 5)(y^2 + 25)\) represents a simplified version of the original polynomial. Mastery of polynomial factorization allows one to navigate complex algebraic expressions with confidence and efficiency.