Problem 59 - Beginning and Intermediate Algebra

Question

Use FOIL to multiply. $$(7 m-3 n)(m-n)$$

Step-by-Step Solution

Answer

The short answer is \((7m - 3n)(m - n) = 7m^2 - 10mn + 3n^2\).

Step 1: Label the terms in each binomial.

We have two binomials: First binomial \((7m - 3n)\): - First term: \(7m\) - Second term: \(-3n\) Second binomial \((m - n)\): - First term: \(m\) - Second term: \(-n\)

Step 2: Apply the FOIL method.

Now, we will multiply the terms using the FOIL method: - First terms: \((7m)(m)\) - Outer terms: \((7m)(-n)\) - Inner terms: \((-3n)(m)\) - Last terms: \((-3n)(-n)\)

Step 3: Multiply the terms.

Multiply the terms found in Step 2: - First terms: \(7m\cdot m = 7m^2\) - Outer terms: \(7m\cdot (-n) = -7mn\) - Inner terms: \(-3n\cdot m=-3mn\) - Last terms: \(-3n\cdot (-n) = 3n^2\)

Step 4: Combine like terms.

There are no like terms in our expression, so we can just add the terms we found in Step 3: \(7m^2 - 7mn - 3mn + 3n^2\)

Step 5: Write the final expression.

The final expression after combining terms is: \(7m^2 - 10mn + 3n^2\) So, \((7m - 3n)(m - n) = 7m^2 - 10mn + 3n^2\).

Key Concepts

Binomial Multiplication

Binomial multiplication refers to the process of multiplying two binomials together to form a polynomial. A binomial is an algebraic expression with two terms, such as

• (7m - 3n)
• (m - n)

To multiply these binomials, we use the FOIL method. FOIL stands for First, Outer, Inner, Last – representing the order in which you multiply the terms of the binomials.
Start with the First terms in each binomial and multiply them. Then move to the Outer terms, followed by the Inner, and finally, multiply the Last terms together. By following this method, it ensures that all terms are accounted for and accurately combined. The result is a polynomial expression.

Polynomial Expression

A polynomial expression is the result obtained from multiplying binomials. It consists of variables raised to different powers and coefficients. The expression often includes several terms combined together, each term being a part of the final expanded form. For example, when multiplying two binomials like

• (7m - 3n)
• (m - n)

the resultant polynomial expression is

• 7m^2 - 10mn + 3n^2

In this polynomial expression, each term (e.g., \(7m^2\), \(-10mn\), and \(+3n^2\)) is a result of the multiplication of terms from the original binomials. This expanded polynomial conveys more information and consists of individually distinguishable components, which are essential in understanding complex algebraic concepts.

Like Terms

Like terms in an algebraic expression are those that have the same variable parts, raised to the same power. In the context of binomial multiplication, identifying and combining like terms are crucial steps to simplifying the resultant polynomial. After deploying the FOIL method, a polynomial expression results,
which may involve like terms. In the given example of multiplying

• (7m - 3n)
• (m - n)

the terms resulting from expansion

• \(-7mn\) and \(-3mn\) are like terms. Combining such terms streamlines the polynomial expression, leading to a simplified form:
  • \(-10mn\)
  
  Being able to identify and simplify like terms enhances your ability to solve and understand algebraic problems efficiently.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. They are fundamental structures in mathematics that represent various relationships and problems. When we discuss algebraic expressions, it encompasses everything from simple variables to more complex polynomial forms.
For instance, consider the expression

• \(7m – 3n\)

.
This is an algebraic expression made up of both constants and variables. It allows us to represent relationships mathematically and find unknown values when solved in equations. Multiplying two algebraic expressions, like binomials, expands them to form more substantive expressions, often resulting in polynomials.
This foundational concept is used across different areas of mathematics, including calculus, physics, and engineering, due to its versatility and capabilities in modeling real-world situations.