Problem 20 - College Algebra

Question

Find each product. $$(x+8)(x+5)$$

Step-by-Step Solution

Answer

The product of the two binomials is \( x^2 + 13x + 40 \).

Step 1: Apply FOIL method - First

Multiply the first terms in each binomial. That gives \( x*x = x^2 \).

Step 2: Apply FOIL method - Outside

Multiply the outside terms in the product. That gives \( x*5=5x \).

Step 3: Apply FOIL method - Inside

Multiply the inside terms in the product. That gives \( 8*x=8x \).

Step 4: Apply FOIL method - Last

Multiply the last terms in each binomial. That gives \( 8*5=40 \).

Step 5: Sum up the results

Add up the results from step 1,2,3 and 4. \( x^2 + 5x + 8x + 40 \).

Step 6: Simplify the expression

Simplify the expression by combining like terms. That gives \( x^2 + 13x + 40 \).

Key Concepts

FOIL Method

The FOIL method is a powerful tool for multiplying two binomials. It's a simple acronym that stands for First, Outside, Inside, and Last. When you multiply two binomials, you apply these four steps in sequence:

• First: Multiply the first terms of each binomial together. For example, in the expression \((x+8)(x+5)\), you first multiply the two \(x\)'s to get \(x^2\).
• Outside: Then, multiply the outer terms, which are the first term of the first binomial and the second term of the second binomial. Continuing with \((x+8)(x+5)\), you get \(x \times 5 = 5x\).
• Inside: Multiply the inner terms, which are the second term of the first binomial and the first term of the second binomial. Here, that's \(8 \times x = 8x\).
• Last: Finally, multiply the last terms of each of the binomials. In the expression we are working with, \(8 \times 5 = 40\).

After going through all these steps, you'll have four terms that you can then combine further. The FOIL method ensures nothing is missed and helps keep everything organized while multiplying binomials.

Binomials

A binomial is a polynomial that contains exactly two terms. Polynomials are algebraic expressions that consist of variables and coefficients, structured in terms of powers. Binomials can take different forms, but they always contain two separate parts. For instance, \((x+8)\) and \((x+5)\) are both binomials. They have the typical form of \(a + b\), where \(a\) and \(b\) can be any numbers or variables.

Understanding binomials is crucial because they are the building blocks for more complex algebraic expressions. They frequently appear in algebra problems, especially when dealing with quadratic equations and factoring. Recognizing binomials and knowing how to manipulate them using techniques like the FOIL method enables you to tackle more intricate problems.

Combining Like Terms

After using the FOIL method to multiply binomials, you will often end up with some similar terms. Combining these like terms is essential for simplifying the expression. Like terms are terms that have the same variables raised to the same powers, even though their coefficients might be different.

For example, after applying FOIL to \((x+8)(x+5)\), we arrive at \(x^2 + 5x + 8x + 40\). Here, \(5x\) and \(8x\) are like terms because they both contain the variable \(x\) raised to the first power. We combine them by adding the coefficients, 5 and 8, to get \(13x\). This results in the simplified expression: \(x^2 + 13x + 40\).

By combining like terms, you not only simplify the expression but also make it easier to interpret and use in further calculations or contexts. Understanding this concept helps to streamline algebraic expressions and resolve equations more efficiently.