Problem 111 - Beginning and Intermediate Algebra

Question

Find the product. $$-3(m-1)^{2}$$

Step-by-Step Solution

Answer

The product of the given expression is \(-3m^2 + 6m - 3\).

Step 1: Expand (m-1)^2

To expand the expression (m-1)^2, we can simply multiply the binomial by itself: \[(m-1)(m-1)\] Using the distributive property (FOIL method) to multiply the binomials, we get: \[m^2 - m - m + 1\] Combining like terms, we arrive at the following expression: \[m^2 - 2m + 1\]

Step 2: Multiply by -3

Now that we have expanded (m-1)^2, we can multiply the expression by -3. Remember to multiply each term by -3: \(-3(m^2 - 2m + 1)\) Applying distributive property: \[-3m^2 + 6m - 3\] The product of the given expression is now fully simplified: \[-3m^2 + 6m - 3\]

Key Concepts

Expanding Binomials

Expanding binomials involves multiplying a two-term expression by itself or another binomial.When you see a binomial like \(m-1\) raised to a power, it means you need to multiply it by itself.This is the first step in expanding. Let's break it down:

• The expression \(m-1\)^2 means \(m-1\) times \(m-1\).
• We write this as \( (m-1)(m-1) \).

Now, you've set the stage to apply another key concept to help with expansion: the distributive property.Expanding binomials fully is crucial as it sets up the expression for further simplification like combining terms and multiplying by constants.

Distributive Property

The distributive property is a fundamental algebraic principle essential for polynomial multiplication. It states that you distribute or multiply each term inside a parenthesis by a term outside the parenthesis. In essence, it follows the formula: \ a(b + c) = ab + ac \.In our case with \( (m-1)(m-1) \), it means applying this property systematically:

• First, multiply the first terms: \(m \times m = m^2\).
• Then, multiply the outer terms: \(m \times -1 = -m\).
• Next, multiply the inner terms: \(-1 \times m = -m\).
• Finally, multiply the last terms: \(-1 \times -1 = 1\).

After distributing through the expression, you collect the products: \[m^2 - m - m + 1\].Using the distributive property is a systematic way and ensures accuracy in polynomial expansion.

Combining Like Terms

After expanding the binomials using the distributive property, you will often need to combine like terms to simplify the expression.Like terms are terms that have the same variable raised to the same power. Here, you're looking for terms that "look alike" so they can be added or subtracted easily.
In the example expression \[m^2 - m - m + 1\], the like terms are \(-m\) and \(-m\).Combining them results in \(-2m\):

• So, \[m^2 - m - m + 1\] simplifies to \[m^2 - 2m + 1\].

This process of combining like terms simplifies the polynomial, making it easier to handle in subsequent operations, such as multiplication by constants.A cleanly simplified expression is crucial for clarity and accuracy in mathematics.