In Exercises 23–48, factor completely, or state that the polynomial is prime.50 - 2y²

Verified step by step guidance

1. Identify the common factor in the polynomial terms. Here, the terms are 50 and \(-2y^2\).
2. Notice that both terms can be divided by 2. So, factor out 2 from the polynomial.
3. Rewrite the polynomial as \(2(25 - y^2)\).
4. Recognize that \(25 - y^2\) is a difference of squares, which can be factored further.
5. Apply the difference of squares formula: \(a^2 - b^2 = (a - b)(a + b)\), where \(a = 5\) and \(b = y\), to factor \(25 - y^2\) into \((5 - y)(5 + y)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This process is essential for simplifying expressions and solving equations. Common techniques include finding the greatest common factor (GCF), using special products like the difference of squares, and applying the quadratic formula when necessary.

Difference of Squares

The difference of squares is a specific factoring pattern that applies to expressions of the form a² - b², which can be factored into (a + b)(a - b). In the given polynomial, 50 - 2y² can be rewritten as 2(25 - y²), which is a difference of squares and can be further factored into 2(5 + y)(5 - y).

Prime Polynomials

A polynomial is considered prime if it cannot be factored into simpler polynomials with integer coefficients. Identifying whether a polynomial is prime is crucial in algebra, as it determines the methods available for solving equations. In this case, recognizing that the polynomial can be factored helps avoid mistakenly labeling it as prime.