Problem 60 - 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: Identify the First, Outer, Inner, and Last terms

First, we need to identify the First (F) term in each expression, which are \(7m\) and \(m\). Next, we identify the Outer (O) terms, which are \(7m\) and \(-n\). For Inner (I) terms, we have \(-3n\) and \(m\), and finally, for the Last (L) terms, we have \(-3n\) and \(-n\).

Step 2: Multiply First, Outer, Inner, and Last terms

Now, we need to multiply each of these pairs of terms: - F: \(7m \times m = 7m^2\) - O: \(7m \times (-n) = -7mn\) - I: \(-3n \times m = -3mn\) - L: \(-3n \times (-n) = 3n^2\)

Step 3: Combine Like Terms

Finally, we will combine any like terms in the sum of these products. In this case, there are no like terms remaining to simplify, so we can write the final answer as: \( (7m - 3n)(m - n) = 7m^2 - 7mn - 3mn + 3n^2 = 7m^2 - 10mn + 3n^2 \)

Key Concepts

Polynomial Multiplication

When we talk about polynomial multiplication, we're discussing how to find the product of two or more polynomial expressions. This concept is an extension of basic multiplication to expressions that are more complex than just numbers.

• Definition: A polynomial is an algebraic expression made up of terms. Each term consists of a constant multiplier and one or more variables raised to a non-negative integer power.
• A product of polynomials is another polynomial that results from multiplying two or more polynomials together.
• When you multiply polynomials, every term from one polynomial has to be multiplied by every term of the other polynomial. For example, to multiply \((7m - 3n)(m - n)\), you have to multiply each term in the first polynomial by each term in the second polynomial.

This process can seem intimidating at first, but it follows logical and consistent steps, like FOIL for binomials or distributing terms for larger polynomials. Mastery of this technique is key in algebra.

Algebraic Expressions

Algebraic expressions form the basis of algebra. They contain numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. Let's break down their components:

• Terms: The individual parts of an expression. In the expression \(7m - 3n\), the terms are \(7m\) and \(-3n\).
• Coefficients and Variables: A coefficient is a numerical or constant factor in front of the variables in a term, like 7 in \(7m\). Variables, such as \(m\) or \(n\), are symbols that represent numbers.
• Operations: Operations in algebraic expressions include addition, subtraction, multiplication, and division. For instance, \(-3n\) implies multiplying \(-3\) by \(n\).

When multiplying algebraic expressions, it's important to follow the rules of arithmetic and apply them to each component carefully. This helps ensure accuracy in your calculations.

Distributive Property

The distributive property is a fundamental principle in algebra that helps simplify expressions and solve equations. Understanding this property will deepen your algebra skills greatly.

• Basic Definition: The distributive property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In symbols, it's expressed as \[a(b+c) = ab + ac\].
• In our example, the FOIL method is a specific application of the distributive property. It breaks down the multiplication of binomials into manageable parts: first, outer, inner, and last terms.
• Applying this property requires you to multiply each term in one polynomial by every term in the other, ensuring no terms are left out. In \((7m - 3n)(m - n)\), we distribute and multiply such that each term from the first binomial interacts with each term in the second binomial.

Understanding and applying the distributive property efficiently leads to success in solving algebraic equations and multiplying complex expressions.