Problem 93 - Intermediate Algebra

Question

Simplify each expression. $$8(2 c d+7 c)-2(c d-3 c)$$

Step-by-Step Solution

Answer

The expression simplifies to \(14cd + 62c\).

Step 1: Distribute 8 into the First Parenthesis

To remove the parentheses, distribute the 8 into each term inside the first set of parentheses: \[ 8(2cd + 7c) = 8 \times 2cd + 8 \times 7c = 16cd + 56c \]

Step 2: Distribute -2 into the Second Parenthesis

Now distribute the -2 into each term inside the second set of parentheses:\[ -2(cd - 3c) = -2 \times cd + (-2) \times (-3c) = -2cd + 6c \]

Step 3: Combine Like Terms

Combine the like terms from the results obtained in previous steps. The expression becomes:\[ 16cd + 56c - 2cd + 6c \] Now, combine \(16cd\) with \(-2cd\) and \(56c\) with \(6c\):\[ (16cd - 2cd) + (56c + 6c) = 14cd + 62c \]

Key Concepts

Distributive Property

The distributive property is a powerful algebraic tool that helps us to multiply a single term by terms inside a parenthesis. You might often encounter expressions with terms grouped by parentheses, and distributing is the process that lets us break those down into simpler parts.

For example, in the expression \(8(2cd + 7c)\), we distribute the 8 across both terms inside the parentheses, meaning we multiply 8 by every term:

• Multiply 8 and \(2cd\) together to get \(16cd\).
• Multiply 8 and \(7c\) to obtain \(56c\).

Thus, we can rewrite our expression as \(16cd + 56c\).

Remember to also be cautious with negative numbers. In our exercise, when distributing \(-2\) across \(cd - 3c\), we must keep track of the negative sign:

• \(-2\) times \(cd\) results in \(-2cd\).
• \(-2\) times \(-3c\) gives us \(+6c\).

Always double-check the signs; they are just as important as the numbers themselves!

Combining Like Terms

Combining like terms is an essential skill in algebra that lets us simplify expressions further. In math, like terms are terms that contain the same variables raised to the same power. Only these terms can be combined.

After using the distributive property, you'll often find yourself with a sum of terms that can be combined. In our example, we ended up with the expression \(16cd + 56c - 2cd + 6c\). Here, we have like terms, \(16cd\) and \(-2cd\), that can be combined as they both contain the same variables, \(cd\).

When combining:

• Add the coefficients of the like terms. Here, the \(cd\) terms are combined as \(16 - 2 = 14\) resulting in \(14cd\).
• Similarly, for the terms \(56c\) and \(6c\), the same principle applies, resulting in \(62c\).

The ultimate simplified expression from our original problem is \(14cd + 62c\).

This process cuts down on complexity, leaving you with a neat and simplified expression. Always look for terms that share variables and degree so you can make these combinations.

Polynomial Expressions

Polynomial expressions are algebraic expressions that involve variables raised to whole-number exponents, which are summed or subtracted. Understanding polynomials is key to many areas of mathematics and they appear frequently in algebra problems.

In our problem, both \(2cd + 7c\) and \(cd - 3c\) are polynomial expressions. They're made up of terms with variables like \(c\) and \(d\), each potentially having different exponents such as \(c\) (which is \(c^1\)) and \(cd\) (which is \(c^1 d^1\)).

• Each term in the polynomial like \(16cd\), \(56c\), etc., is called a monomial.
• When the polynomial consists of two terms, it is specifically known as a binomial, and when there are three, it's a trinomial.

When simplifying polynomials, always maintain awareness of the degree of each term, which is the sum of the powers of all variables in that term.

In our simplified expression \(14cd + 62c\), both terms are actually monomials within a larger polynomial expression. This simplicity is the power of understanding and manipulating polynomial expressions!