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In mathematics, a common technique in factoring polynomials is the **difference of squares**. This technique applies when you have a polynomial of the form \(a^2 - b^2\). The principle behind it is that any expression structured as a difference of two squares can be factored into the product of two binomials:

• \((a - b)(a + b)\)

This approach simplifies expressions by breaking them down into more manageable parts. Let's apply this to our exercise with \(T^2 - 100\).

We see that \(T^2\) is a perfect square (specifically \(a^2\)), and \(100\) is another perfect square (\(b^2\) since \(100 = 10^2\)). Therefore, \(T^2 - 100\) can be rewritten as:

• \((T - 10)(T + 10)\)

This is a standard case of the difference of squares, making the polynomial easier to work with.

A **binomial** is a polynomial that consists of exactly two terms. It's often simpler to manipulate and factor than polynomials with more terms. In our case, each factor \(T - 10\) and \(T + 10\) is a binomial. When multiplying binomials together, we apply the distributive property, which involves multiplying each term in the first binomial by each term in the second binomial.

The special nature of our expression consisting of difference of squares simplifies this further since:

• \((T - 10)(T + 10)\) alone represents the difference of squares.
• When squared, it becomes \(((T - 10)(T + 10))^2\).

With this expression, you can see how multiplication of binomials is used to factor and further simplify polynomials.

Polynomial simplification involves reducing a polynomial to its simplest form. The process typically includes factoring, combining like terms, and sometimes even distributing terms. For this particular problem, our simplification began with recognizing the difference of squares, which allowed us to factor \(T^2 - 100\) into easier binomials.

Once we input this factorization into our original expression, \((T^2 - 100)(T - 10)(T + 10)\), it became simpler to group and manipulate these factors. After identifying the repeating binomials, we simplified it further to \((T - 10)(T + 10)(T - 10)(T + 10)\), and ultimately to a compact form: \(((T - 10)(T + 10))^2\).

• This is considered the simplest form since it is concise and contains the minimum necessary components to represent the expression.
• Simplification helps to better understand and solve polynomial expressions in mathematics, offering clarity in analysis and computation.