Problem 84 - Algebra A Combined Function

Question

Multiply. $$ (x-5)(x+10) $$

Step-by-Step Solution

Answer

The product is \(x^2 + 5x - 50\).

Step 1: Identify the Binomials

We are given two binomials to multiply: \((x - 5)\) and \((x + 10)\). Our task is to find their product by expanding the expression.

Step 2: Apply the Distributive Property

We will use the distributive property, also known as the FOIL method (First, Outer, Inner, Last), to multiply each term in the first binomial by each term in the second binomial.

Step 3: Multiply First Terms

Multiply the first terms of each binomial: \(x \times x = x^2\).

Step 4: Multiply Outer Terms

Multiply the outer terms: \(x \times 10 = 10x\).

Step 5: Multiply Inner Terms

Multiply the inner terms: \(-5 \times x = -5x\).

Step 6: Multiply Last Terms

Multiply the last terms: \(-5 \times 10 = -50\).

Step 7: Combine Like Terms

Combine all the products: \(x^2 + 10x - 5x - 50\). Simplify by combining like terms: \(10x - 5x = 5x\). Thus, the expanded expression is \(x^2 + 5x - 50\).

Key Concepts

Binomials

Binomials are expressions that consist of two terms. In this exercise, the two binomials given are

• \((x - 5)\)
• \((x + 10)\)

Each binomial is composed of terms separated by a plus or minus sign. Here:

• In the binomial \((x - 5)\), \(x\) is one term, and \(-5\) is the other.
• In \((x + 10)\), \(x\) is a term, and \(10\) is the other term.

Understanding that we are dealing with binomials is critical because it means we are going to work with these two-part expressions.
Recognizing and identifying binomials sets the stage for applying multiplication techniques such as the distributive property.

Distributive Property

The distributive property is a valuable tool in algebra that helps us to simplify expressions. When applied, it involves multiplying each term inside a parenthesis by another term outside the parenthesis.
For two binomials, the distributive property is applied as follows:

• First, multiply each term in the first binomial with each term in the second binomial.
• Here, you will pair each element from the first binomial and distribute it to each element from the second binomial.

This method ensures that all combinations of terms are covered, and no part of the expression is left out.
The distributive property guides us in systematically expanding binomials, setting up for combining like terms later.

FOIL Method

The FOIL method is a specific application of the distributive property for multiplying two binomials. FOIL stands for First, Outer, Inner, Last.

• First: Multiply the first terms in each binomial. In this case, \(x \times x = x^2\).
• Outer: Multiply the outer terms. Here, \(x \times 10 = 10x\).
• Inner: Multiply the inner terms. That would be \(-5 \times x = -5x\).
• Last: Multiply the last terms. So, \(-5 \times 10 = -50\).

The FOIL method provides a clear structure for binomial multiplication, making it easier to remember and execute.
After calculating each component, combine them: \(x^2 + 10x - 5x - 50\).
Remember to simplify by combining like terms. The result is the expanded and combined form of the expression: \(x^2 + 5x - 50\).