Problem 44 - Elementary and Intermediate Algebra

Question

Multiply. See Examples 4 and 5. $$ (n+8)(n-10) $$

Step-by-Step Solution

Answer

The product is \( n^2 - 2n - 80 \).

Step 1: Use the Distributive Property

We will use the distributive property (also known as the FOIL method for binomials) to multiply the two binomials: \( (n+8)(n-10) \). This property involves multiplying each term in the first binomial by every term in the second binomial. This means we will perform four multiplications: multiply first terms, outer terms, inner terms, and last terms.

Step 2: Multiply the First Terms

Multiply the first term in each binomial: \( n \times n = n^2 \).

Step 3: Multiply the Outer Terms

Multiply the outer terms of the binomials: \( n \times -10 = -10n \).

Step 4: Multiply the Inner Terms

Multiply the inner terms of the binomials: \( 8 \times n = 8n \).

Step 5: Multiply the Last Terms

Multiply the last term in each binomial: \( 8 \times -10 = -80 \).

Step 6: Combine the Products

Combine all the products obtained from Steps 2, 3, 4, and 5. This gives us: \( n^2 - 10n + 8n - 80 \).

Step 7: Simplify the Expression

Combine the like terms \(-10n + 8n = -2n\). Hence, the simplified expression is: \( n^2 - 2n - 80 \).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term by a group of terms within parentheses. It simplifies expressions by distributing the multiplication over addition or subtraction inside the parentheses. This is particularly useful when dealing with polynomials. For example, if you have an expression like \[a(b + c)\]this can be expanded to \[ab + ac\].In the case of the exercise, the distributive property is used in the FOIL process to handle multiplication of two binomials, where each term in one binomial is multiplied by each term in the other. This breaks a potentially complex expression into simpler, manageable parts.

Binomials

Binomials are algebraic expressions that contain exactly two terms. These terms are separated by either a plus or minus sign. For instance, in the expression \[(n + 8),\]\[(n - 10)\]both are binomials. Binomials appear frequently in algebraic problems and are the building blocks for polynomial expressions. Understanding binomials is essential because they help form equations that model real-world situations. When multiplying binomials, as in the exercise, the result is typically a trinomial or higher degree polynomial, which requires careful simplification through combining like terms.

FOIL Method

The FOIL method is a specific application of the distributive property, particularly for multiplying two binomials. FOIL stands for:

• First: Multiply the first terms of each binomial.
• Outer: Multiply the outer terms of the expression.
• Inner: Multiply the inner terms.
• Last: Multiply the last terms of each binomial.

For the given exercise \[(n+8)(n-10),\]the steps are:

• First: \(n \times n = n^2\)
• Outer: \(n \times -10 = -10n\)
• Inner: \(8 \times n = 8n\)
• Last: \(8 \times -10 = -80\)

These products are then combined and simplified, showcasing how each pair of terms is handled systematically to result in a quadratic expression.

Polynomial Multiplication

Polynomial multiplication is the process of multiplying two polynomials together to form a new polynomial. This involves taking each term in the first polynomial and multiplying it by each term in the second polynomial. It is a versatile tool in algebra, enabling the expansion of expressions, which is crucial in solving equations and simplifying expressions.In the context of the exercise, polynomial multiplication is applied to the two binomials, \[(n+8)(n-10).\]By applying the FOIL method, the result is \[n^2 - 2n - 80,\]which is a quadratic polynomial. Simplifying these terms often involves combining like terms and rearranging the expression to standard polynomial form, which makes it easier to analyze, solve, or graph.