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The square of a binomial is a frequent operation in algebra that results in expanding an expression like \((x-4)^2\). When a binomial is squared, it means multiplying the binomial by itself: \((x-4) \times (x-4)\). While you could multiply out the terms directly, it’s often simpler to use the square of a binomial formula:

• \((a-b)^2 = a^2 - 2ab + b^2\)

In this case, \(a\) corresponds to \(x\), and \(b\) to \(4\). Substituting these into the formula, the structure becomes \(x^2 - 2(x)(4) + 4^2\). This expansion streamlines the multiplication process and helps avoid small mistakes in calculations.
Utilizing this method provides a clear path to reach the solution, ensuring that all terms are correctly accounted for.

Once you have used the formula to expand the binomial, the next step is simplifying the expression. Simplification involves combining like terms and performing basic arithmetic to reduce the expression into its simplest form. For the expression \(x^2 - 2(x)(4) + 4^2\), each part can be simplified:

• \(2(x)(4)\) calculates to be \(8x\)
• \(4^2\) evaluates to \(16\)

Thus, the expression \(x^2 - 8x + 16\) is obtained. This final form is much simpler to work with and provides a neat, straightforward representation of the original expanded expression. Effective simplification ensures clarity and enhances your ability to handle more complex algebraic tasks confidently.

In algebra, polynomial multiplication isn't solely limited to basic terms; it often involves larger expressions like binomials. Understanding how to multiply binomials is essential since it appears in various contexts, including expanding binomial squares.
The basic idea of polynomial multiplication involves multiplying each term in one polynomial by each term in another and then summing up the results. For the square of a binomial such as \((x-4)^2\), multiplying \((x-4)\) by itself requires careful distribution of each term:

• First, multiply the first terms: \(x \times x = x^2\)
• Then, the outer terms: \(x \times (-4)\)
• Next, the inner terms: \(-4 \times x\)
• And finally, the last terms: \(-4 \times -4 = 16\)

Collecting these results gives you the expanded polynomial, which in the case of a squared binomial, usually aligns with the simplified expression from applying the binomial formula. This ensures that your resulting expression covers all products without missing any interactions between terms.