Question

in a short answer, explain if \\(9x^2 - 25x\\) can be factored using difference of squares. be detailed in your response (upload your hand written out work to provide proof of your response)

Explanation:

Identify the expression and the factoring method

The given expression is \(9x^2 - 25x\). We need to determine if it can be factored using the Difference of Squares method.

Define the difference of squares pattern

A binomial can be factored as a difference of squares if it fits the algebraic form:

$$a^2 - b^2 = (a - b)(a + b)$$

This requires both terms to be perfect squares separated by a subtraction sign.

Analyze the terms of the expression

Let's examine each term in \(9x^2 - 25x\):

• The first term is \(9x^2\), which is a perfect square: \((3x)^2\).
• The subtraction sign is present between the terms.
• The second term is \(25x\). While the coefficient \(25\) is a perfect square \((5^2)\), the variable part \(x\) is raised to the power of \(1\), which is not an even power and thus not a perfect square.

Determine factorability using other methods

Using Polynomial Factoring, we can factor the expression by finding the greatest common factor (GCF) instead:

$$9x^2 - 25x = x(9x - 25)$$

This confirms that the expression is factorable, but not by using the difference of squares.

Answer:

No, the expression \(9x^2 - 25x\) cannot be factored using the difference of squares.

While the first term \(9x^2\) is a perfect square \((3x)^2\) and there is a subtraction sign, the second term \(25x\) is not a perfect square because the variable \(x\) has an exponent of \(1\). To be factored as a difference of squares, both terms must be perfect squares (e.g., \(9x^2 - 25\)). Instead, this expression can only be factored by taking out the greatest common factor, \(x\), resulting in \(x(9x - 25)\).