How can I determine if a polynomial is a difference of two squares, specifically for the polynomials x² – 14, x² – 14, x² – 49, and x² – 49?

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To determine whether a polynomial is a difference of two squares, you can use the mathematical identity:

a² – b² = (a – b)(a + b)

In this case, a² represents the first term, and b² represents the second term. A polynomial can be classified as a difference of two squares if it can be expressed in the form a² – b².

Let’s analyze the polynomials provided in your question:

  1. x² – 14: Here, we can see that this expression does not fit the difference of two squares format because 14 is not a perfect square (√14 is not an integer). Hence, x² – 14 is not a difference of two squares.
  2. x² – 14 (again): Just like the one before, since it is the same expression, x² – 14 is not a difference of two squares.
  3. x² – 49: Now, let’s examine this polynomial. Here we have:
    a = x (since (x)² = x²)
    b = 7 (since 7² = 49)
    Therefore, we can rewrite this as:

    x² – 7² = (x – 7)(x + 7) which confirms that x² – 49 is a difference of two squares.

  4. x² – 49 (again): Similar to this one. As previously determined, x² – 49 is a difference of two squares.

In conclusion, out of the given polynomials, only x² – 49 qualifies as a difference of two squares. Remember, for a polynomial to fit this classification, the terms must each be perfect squares, indicating a valid factorization into the product of two binomials.

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