Problem 48 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the polynomials. \((9 m+4 n-1)(2 m+8)\)

Step-by-Step Solution

Answer

The product is \(18m^2 + 70m + 8nm + 32n - 8\).

Step 1: Distribute Each Term

To multiply the polynomials \((9m + 4n - 1)(2m + 8)\), use the distributive property. Multiply each term in the first polynomial by every term in the second polynomial, one at a time:- First, multiply \(9m\) by \(2m\) and add it to \(9m\) times \(8\).- Then multiply \(4n\) by \(2m\) and add it to \(4n\) times \(8\).- Lastly, multiply \(-1\) by \(2m\) and add it to \(-1\) times \(8\).

Step 2: Calculate Each Product

Now, perform each multiplication from Step 1:- \(9m \times 2m = 18m^2\)- \(9m \times 8 = 72m\)- \(4n \times 2m = 8nm\)- \(4n \times 8 = 32n\)- \(-1 \times 2m = -2m\)- \(-1 \times 8 = -8\)

Step 3: Combine Like Terms

Add all the calculated terms together and then combine like terms:\[18m^2 + 72m + 8nm + 32n - 2m - 8\]Combine the \(m\) terms:- \(72m - 2m = 70m\)Thus, the simplified expression is:\[18m^2 + 70m + 8nm + 32n - 8\]

Step 4: Present the Simplified Expression

The final expression, after combining like terms, is:\[18m^2 + 70m + 8nm + 32n - 8\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra, and it plays a crucial role in polynomial multiplication. It allows you to multiply a single term by two or more terms that are added or subtracted inside parentheses. The property states that: For any numbers or expressions, the equation: \( a(b+c) = ab + ac \)can be used.In the context of multiplying polynomials, this property helps you distribute each term of one polynomial to every term of the other polynomial:

• For the polynomial \((9m + 4n - 1)(2m + 8)\), distribute every term in \(9m + 4n - 1\) to \(2m + 8\).
• This means multiplying: \(9m\) with \(2m\) and \(9m\) with \(8\), then \(4n\) with \(2m\) and \(4n\) with \(8\), and so on.

Using the distributive property ensures you multiply each pair of terms once, which is essential for correctly expanding polynomial expressions. Each operation is done separately before combining the results.

Combining Like Terms

Combining like terms is an essential skill when working with polynomial expressions. It involves simplifying expressions by adding or subtracting terms that have the same variables raised to the same powers. Essentially, you can think of 'like terms' as terms in a polynomial that share the same "base".To effectively combine like terms:

• Identify terms that have identical variable parts. For example, \(72m\) and \(-2m\) both have the variable \(m\).
• Add or subtract the coefficients while keeping the variable part unchanged. In our case, \(72m - 2m = 70m\).

This process reduces the number of terms, simplifying your expressions and making them easier to interpret and use. In this exercise, we focus on combining the \(m\) terms to arrive at the simplified polynomial: \(18m^2 + 70m + 8nm + 32n - 8\). This step is critical in getting to the final answer efficiently.

Polynomial Expressions

Polynomial expressions are algebraic expressions made up of terms, which can include variables, coefficients, and constants. These terms are joined by addition or subtraction. The highest power of the variable(s) in the expression determines its degree.A few key points about polynomial expressions:

• A polynomial like \(9m + 4n - 1\) with another \(2m + 8\) can be multiplied to form a new polynomial.
• Each product of terms from separate polynomials generates a new term for the resulting polynomial.
• The structure of polynomials allows them to model a variety of real-world situations, making them versatile and essential in mathematics.

Understanding polynomial expressions involves recognizing how each term contributes to the overall structure. By using the distributive property and combining like terms, you can produce a simplified version of a complex polynomial, such as \(18m^2 + 70m + 8nm + 32n - 8\). These operations help you manipulate polynomials for solving equations and understanding function behavior.