How do you determine if a polynomial is the difference of two squares?
are there two terms?
are both terms perfect squares?
Is there a minus sign between them?
e.g. 16x^2 - 25y^2 ---- YES
2x^ - 16 ---- no
x^ + y^2 --- no
Thanks a bunch Reiny!
To determine if a polynomial is the difference of two squares, follow these steps:
1. Identify the polynomial: Write down the given polynomial in its standard form, typically in decreasing order of the exponent.
2. Check if it is a binomial: Make sure the polynomial has only two terms. If there are more or fewer than two terms, it cannot be the difference of two squares.
3. Verify the exponent: Check if both terms have even exponents. If any of the terms has an odd exponent, it cannot be the difference of two squares.
4. Decompose the terms: Factor each term using the difference of squares formula:
a^2 - b^2 = (a + b)(a - b)
5. Compare with the given polynomial: Compare the factored terms you obtained in step 4 with the given polynomial. If the polynomial matches the pattern (a + b)(a - b), then it is the difference of two squares.
Here's an example to illustrate the process:
Given polynomial: x^2 - 9
Step 1: Identify the polynomial: x^2 - 9.
Step 2: Check if it is a binomial: Yes, it has only two terms.
Step 3: Verify the exponent: Both terms have even exponents (2 and 0).
Step 4: Decompose the terms: Using the difference of squares formula, we have:
x^2 - 9 = (x + 3)(x - 3)
Step 5: Compare with the given polynomial: We see that the factored terms (x + 3)(x - 3) match the given polynomial x^2 - 9. Therefore, x^2 - 9 is the difference of two squares.
Overall, to determine if a polynomial is the difference of two squares, check if it is a binomial with both terms having even exponents. Factor the terms using the difference of squares formula and compare the results to the given polynomial.