Problem 75 - Elementary and Intermediate Algebra

Question

Perform the operations. $$ (m+10)^{2}-(m-8)^{2} $$

Step-by-Step Solution

Answer

The result of the operation is \(36m + 36\).

Step 1: Expand the first square

The expression \((m+10)^2\) can be expanded using the identity \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = m\) and \(b = 10\). So, \((m+10)^2 = m^2 + 2 \times m \times 10 + 10^2\), which simplifies to \(m^2 + 20m + 100\).

Step 2: Expand the second square

Use the identity for expanding a square, \((a-b)^2 = a^2 - 2ab + b^2\), to expand \((m-8)^2\). In this case, \(a = m\) and \(b = 8\). Thus, \((m-8)^2 = m^2 - 2 \times m \times 8 + 8^2\), which simplifies to \(m^2 - 16m + 64\).

Step 3: Subtract the expanded squares

Now substitute the expanded forms obtained in Steps 1 and 2 into the original expression: \((m+10)^2 - (m-8)^2\) becomes \((m^2 + 20m + 100) - (m^2 - 16m + 64)\). Subtract each term of the second expansion from the first, yielding \(m^2 - m^2 + 20m + 16m + 100 - 64\).

Step 4: Simplify the expression

Simplify the result from Step 3: \(m^2 - m^2\) cancels out, and \(20m + 16m = 36m\). The constants combine as \(100 - 64 = 36\). Therefore, the simplified expression is \(36m + 36\).

Key Concepts

Polynomial Expansion

Polynomial expansion refers to expressing an algebraic equation in an extended form by applying certain mathematical rules. In the context of expressions like

• given a polynomial, we aim to re-write it into a format that makes operations like addition, subtraction, or even graphing easier.
• In the problem e.g., the expression \((m+10)^2\), we utilize the identity \((a+b)^2 = a^2 + 2ab + b^2\).

This describes how to expand a binomial into a trinomial by expanding each term accordingly.
The process extends to any polynomial expression, translating a concise expression into a sum of terms. For instance, when we expand \((m+10)^2\), we substitute:

• \(a = m\)
• \(b = 10\)
• Then compute \(a^2 + 2ab + b^2\)

To get: \(m^2 + 20m + 100\).Similarly, the expansion of \((m-8)^2\) results in:\(m^2 - 16m + 64\).By expanding each polynomial, we can perform further operations like subtraction or simplification. This step is crucial for understanding more complicated algebraic tasks.

Binomial Theorem

The Binomial Theorem is an essential mathematical principle that allows us to expand expressions raised to a power.It specifically helps in simplifying expressions like \((m+10)^2\) or other similar binomials without manually multiplying. The theorem states that \((a+b)^n\) can be expanded using

• the expression: \( \sum_{k=0}^{n} \binom{n}{k} a^{n-k}b^{k} \).
• This simplifies the expansion for any power \(n\).

In simpler terms, it generates the coefficients and the terms needed to form a polynomial from a binomial. For most basic operations, one is likely to use the special cases such as

• \((a+b)^2 = a^2 + 2ab + b^2\)
• which saves time and reduces errors in calculations.

In the given exercise:

• The theorem transforms the binomials into trinomial forms so they can easily be managed in further steps.

In practice, while most exercises or simple power expansions don't require the full theorem, understanding it gives a strong foundation in algebra and problem-solving.

Difference of Squares

The difference of squares is a special algebraic pattern used to simplify expressions where two squared terms are subtracted.When you see an expression like \((a+b)^2 - (a-b)^2\), it's possible to rewrite it as a product of a sum and a difference:

• \((a + b)^2 - (a - b)^2 = a^2 + 2ab + b^2 - (a^2 - 2ab + b^2)\)
• This simplifies into \(4ab\).

In our exercise,

• the problem \((m+10)^2 - (m-8)^2\) uses this idea.

By expanding both squares separately and subtracting, we recognize the simplification - the middle terms, those related to \(a^2\), cancel out, leaving only the term involving both binomials \(36m + 36\).Using the difference of squares saves time and aids in solving a variety of algebra problems more efficiently. It's an invaluable tool for quick simplification and demonstrating algebraic elegance and efficiency.