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When working with expressions and equations, the distributive property is a fundamental concept. It asserts that multiplying a sum by a number yields the same result as multiplying each addend by the number and then summing up the products. This can be stated as:
\[a(b + c) = ab + ac\]
In the example given, we start by applying the distributive property after expanding the squared term. Applying the squared term of a binomial means we use the formula:
\[(a - b)^2 = a^2 - 2ab + b^2\]
We set \(a = 4r\) and \(b = 2\). This step simplifies applying the formula correctly to get:
\[(4r - 2)^2 = (4r)^2 - 2(4r)(2) + 2^2\]. This is an outcome of applying the distributive property.

Expanding polynomials involves the expression of a polynomial, typically a binomial, in its general form. This allows you to work with and solve the polynomial more easily. For instance:
When expanding \((4r - 2)^2\), a common formula is employed which is:
\[(a - b)^2 = a^2 - 2ab + b^2\]
Let's break this down:

• First, find \((4r)^2\) which equals \(16r^2\)
• Next, find \(-2(4r)(2)\) which gives \(-16r\)
• Finally, calculate \(2^2\) which is \(4\)

Upon combining these terms, we get the polynomial: \(16r^2 - 16r + 4\). This expanded polynomial makes it easier to apply further operations or solve equations.

A commonly tricky part of expanding binomials is correctly distributing a leading negative sign across the terms inside the expression. Here is how you do it effectively:
Given the expanded polynomial \(16r^2 - 16r + 4\) and a leading negative sign, namely,
:\[-(4r-2)^2\]
Distributing this negative sign across the terms inside gives:

• \(-16r^2\)
• \(+16r\)
• \(-4\)

Each term's sign changes in the polynomial as a result of the negative distribution. Therefore, the final answer is: \(-16r^2 + 16r - 4\). This correct handling of negative signs prevents errors in calculations and simplifies solving polynomial equations.
Getting comfortable with distributing a negative sign is crucial for more complex algebra problems.