Question

select all the correct answers.

which expressions can be factored using the difference of squares identity?

\\(6x^2 - 81\\)

\\(49x - x^4\\)

\\(32y^2 - 8z^2\\)

\\(x^4 - 400\\)

\\(4a^2 - 64b^5\\)

Explanation:

Analyze the difference of squares criteria

An expression can be factored using the difference of squares identity \(A^2 - B^2 = (A - B)(A + B)\) if it is a binomial subtraction where both terms can be written as perfect squares (with rational coefficients and integer exponents).

Evaluate each given expression

• \(6x^2 - 81\): The coefficient \(6\) is not a perfect square integer.
• \(49x - x^4\): The term \(49x\) has an odd power of \(x\), so it is not a perfect square.
• \(32y^2 - 8z^2\): Factoring out the greatest common factor (GCF) of \(8\) yields \(8(4y^2 - z^2)\). The binomial inside, \(4y^2 - z^2\), is a difference of squares: \((2y)^2 - z^2\).
• \(x^4 - 400\): This can be written as \((x^2)^2 - 20^2\), which is a difference of squares.
• \(4a^2 - 64b^5\): The term \(64b^5\) has an odd exponent on \(b\), so it is not a perfect square.

Identify the correct options

The expressions that can be factored using the difference of squares identity are:

• \(32y^2 - 8z^2\)
• \(x^4 - 400\)

Answer:

• \(6x^2 - 81\)
• \(49x - x^4\)
• \(32y^2 - 8z^2\) (Correct answer)
• \(x^4 - 400\) (Correct answer)
• \(4a^2 - 64b^5\)