Problem 91 - Elementary Algebra

Question

Simplify the algebraic expressions for the following problems. $$ (8 b+2 c)(2 b-c) $$

Step-by-Step Solution

Answer

Question: Simplify the algebraic expression \((8b + 2c)(2b - c)\) Answer: The simplified expression is $$16b^2 - 4bc - 2c^2$$.

Step 1: Identify the given expression

The given expression is \((8b + 2c)(2b - c)\).

Step 2: Apply the distributive property

Using the distributive property (also known as the FOIL method), we'll multiply each term in the first binomial with each term in the second binomial. So we will have \((8b)(2b) + (8b)(-c) + (2c)(2b) + (2c)(-c)\).

Step 3: Multiply the terms

Now, multiply the terms together: \((8b)(2b) = 16b^2\), \((8b)(-c) = -8bc\), \((2c)(2b) = 4bc\), and \((2c)(-c) = -2c^2\). Now, we have the expression \(16b^2 - 8bc + 4bc - 2c^2\).

Step 4: Combine like terms

Next, we will combine the like terms in the expression. The like terms are \(-8bc\) and \(4bc\): \(16b^2 - 8bc + 4bc - 2c^2 = 16b^2 - 4bc - 2c^2\).

Step 5: Write the simplified expression

Finally, the simplified expression for the given algebraic expression is: $$16b^2 - 4bc - 2c^2$$

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra used to simplify expressions and solve equations. It involves multiplying each term inside a set of parentheses by a factor outside of them. This is especially useful when dealing with binomials, like in the example expression \((8b + 2c)(2b - c)\).To apply the distributive property:

• Multiply each term in the first binomial with each term in the second binomial.
• This is also known as the FOIL method, which stands for First, Outer, Inner, and Last terms.

In the expression \((8b + 2c)(2b - c)\), you'll expand it using the distributive property to get:\( (8b)(2b) + (8b)(-c) + (2c)(2b) + (2c)(-c) \).Each multiplication step expands the original expression, breaking it into smaller parts that are simpler to manage.

Binomials

Binomials are algebraic expressions that contain exactly two terms. These terms are usually separated by either a plus or a minus sign. In our problem, \(8b + 2c\) and \(2b - c\) are examples of binomials. Understanding how to work with binomials is crucial for expanding expressions and applying algebraic properties like the distributive property.Key points about binomials:

• Binomials can be added, subtracted, multiplied, and divided, just like any other algebraic expressions.
• When multiplying binomials, every term from the first binomial must be multiplied by every term in the second binomial.
• This often results in a quadratic expression with like terms that need combining.

In algebra, mastering binomials can simplify understanding more complex problems by breaking them down into manageable parts.

Like Terms

In algebra, like terms are terms that have identical variable parts, although their coefficients can differ. Like terms allow us to simplify expressions by combining them. For instance, in the expression \(16b^2 - 8bc + 4bc - 2c^2\), the terms \(-8bc\) and \(4bc\) are like terms because they both contain the variable part \(bc\).To combine like terms:

• Identify terms with the same variable and power.
• Add or subtract the coefficients of these terms.
• Keep the variable part unchanged.

Applying this to our example, \(-8bc + 4bc = -4bc\), resulting in the simplified expression \(16b^2 - 4bc - 2c^2\).By recognizing and combining like terms, we can greatly simplify algebraic expressions, making them easier to interpret and solve.