Problem 22 - Beginning and Intermediate Algebra

Question

Perform the indicated operations and simplify. $$6\left(7 m^{2}+7 m+9\right)-11\left(4 m^{2}+7 m+1\right)$$

Step-by-Step Solution

Answer

The simplified expression for the given problem is: \(-2m^2 - 35m + 43\).

Step 1: Distribute the constants

First, distribute the constants 6 and -11 to the respective terms inside the parentheses: \(6(7m^2 + 7m + 9) - 11(4m^2 + 7m + 1)\) \(42m^2 + 42m + 54 - 44m^2 - 77m - 11\) Next, combine like terms.

Step 2: Combine like terms

We will now combine the like terms (terms with the same power of the variable): \(42m^2 - 44m^2\) for the squared terms \(42m - 77m\) for the m terms \(54 - 11\) for the constant terms So the expression becomes: \(-2m^2 - 35m + 43\) Now we have the simplified expression.

Key Concepts

Polynomial Operations

Polynomial operations are the set of basic arithmetic operations performed on polynomials. A polynomial is a mathematical expression that can have constants, variables, and exponents. Understanding how to manipulate these expressions is crucial in algebra. In this exercise, we have two polynomials within parentheses, each being multiplied by a constant. This problem requires us to perform two particular polynomial operations: distribution and combining like terms. Polynomial operations often include:

• Addition, where terms from different polynomials are added together.
• Subtraction, where terms are subtracted from one another.
• Multiplication, which might include distributing constants or multiplying terms across two polynomials.

Once these operations are completed, the polynomial should be in its simplest form, combining all terms that share the same degree.

Distribution

Distribution is a fundamental operation in algebra that involves multiplying each term inside an expression by a factor outside the parentheses. In the given exercise, we apply the distribution method twice. Firstly, with the constant 6 and then with the constant -11.When you distribute, remember to:

• Multiply each term within the parentheses by the constant outside.
• Pay attention to signs, as they change when you're distributing a negative constant.

For this problem, distributing 6 across the terms within the first parentheses results in: \(6 \times 7m^2 + 6 \times 7m + 6 \times 9 = 42m^2 + 42m + 54\).Similarly, distributing -11 across the terms in the second set of parentheses gives:\(-11 \times 4m^2 - 11 \times 7m - 11 \times 1 = -44m^2 - 77m - 11\). Distribution ensures that all terms in an expression are appropriately accounted for and set up for further simplification by combining like terms.

Combining Like Terms

Combining like terms is a technique used to simplify expressions by merging terms with the same variable and exponent. After distribution in the given polynomial problem, combining like terms simplifies the expression by reducing it to fewer terms.For our specific problem, the terms were:- \(42m^2\) and \(-44m^2\): These are like terms because they both contain the variable \(m^2\). Combine them to get \(-2m^2\).- \(42m\) and \(-77m\): Both are linear terms with the variable \(m\). Their combination results in \(-35m\).- \(54\) and \(-11\): These are constant terms. Adding them together yields \(43\).By combining like terms, the expression is simplified into \(-2m^2 - 35m + 43\). This step is crucial as it condenses the polynomial to its simplest form.