Problem 110 - Beginning and Intermediate Algebra

Question

Find the product. $$-5 c(c+4)^{2}$$

Step-by-Step Solution

Answer

The product of the given expression is \(-5c^3 - 40c^2 - 80c\).

Step 1: Identify the terms and their power

In the given expression \(-5c(c+4)^2\), we have a constant term \(-5\), a variable term \(c\), and a binomial term \((c+4)^2\).

Step 2: Expand the binomial term

First, we will expand the binomial term \((c+4)^2\). This can be done using the formula: \((a+b)^2 = a^2 + 2ab + b^2\), with \(a=c\) and \(b=4\). Expanding the binomial term using the formula: \((c+4)^2 = c^2 + 2(c)(4) + 4^2 = c^2 + 8c + 16\)

Step 3: Multiply each term by the constant and variable terms

Now we will multiply each term of the expanded binomial by the constant term \(-5\) and the variable term \(c\): \(-5c(c^2) + (-5c)(8c) + (-5c)(16)\) This simplifies to: \(-5c^3 - 40c^2 - 80c\)

Step 4: Write the final expression

The final expression after expanding and simplifying is: \(-5c^3 - 40c^2 - 80c\) The product of the given expression is \(-5c^3 - 40c^2 - 80c\).

Key Concepts

Binomial Expansion

Binomial expansion is a method used to express the power of a binomial as a sum of terms involving the coefficients of the expansion. The binomial theorem provides a systematic way of expanding expressions that are raised to any power without multiplying each term. Here, it's crucial to remember the formula:

• \((a+b)^2 = a^2 + 2ab + b^2\)

To apply this, identify \(a\) and \(b\) in your binomial, which in the exercise is \(c\) and \(4\). Applying the formula, we find that \((c+4)^2 = c^2 + 2(c)(4) + 4^2\). This results in:

• \(c^2 + 8c + 16\)

This expanded form is easier to manage when you need to multiply or further manipulate the expression in algebraic operations.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. They are foundational in algebra for representing real-world situations and in forming equations or formulas.

• For example, in the exercise, the given expression is \(-5c(c+4)^2\).

This can be broken down into parts:

• \(-5\) is a constant.
• \(c\) is a variable.
• \((c+4)^2\) is a binomial term, which we expand to simplify calculations.

Understanding how to manipulate algebraic expressions—including expanding binomials, distributing coefficients, and combining like terms—is crucial for solving many types of math problems. This skill allows you to move from complex forms to simpler, more manageable expressions.

Monomials

Monomials are algebraic expressions with only one term, which can include numbers, variables, or both, multiplied together. They are the building blocks of larger algebraic expressions.

• For instance, in the solution, the term \(-5c^3\) is a monomial.
• It comprises a product of the constant \(-5\) and the variable \(c\) raised to the power of three.

Understanding monomials is important because when you multiply, divide, or perform operations with polynomials (which contain multiple monomials), handling each monomial correctly ensures the whole expression is simplified correctly. In polynomial multiplication, such as in our exercise, each expanded term like \(-5c(c^2)\) results in a monomial that contributes to the final polynomial product. Recognizing and combining such terms properly leads to an accurate and simplified result.