Problem 32 - Algebra A Combined Function

Question

Multiply. \((x+2)(x+9)\)

Step-by-Step Solution

Answer

The product is \(x^2 + 11x + 18\).

Step 1: Understand the Problem

We need to multiply the binomials \((x+2)(x+9)\). This involves expanding the expression using distribution or the FOIL method.

Step 2: Apply the FOIL Method

The FOIL method stands for First, Outer, Inner, Last. We will multiply each term in the first binomial \((x + 2)\) by each term in the second binomial \((x + 9)\).

Step 3: Multiply the First Terms

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

Step 4: Multiply the Outer Terms

Multiply the outer terms: \(x \times 9 = 9x\).

Step 5: Multiply the Inner Terms

Multiply the inner terms: \(2 \times x = 2x\).

Step 6: Multiply the Last Terms

Multiply the last terms: \(2 \times 9 = 18\).

Step 7: Combine All Products

Add all the products obtained from steps 3 to 6: \(x^2 + 9x + 2x + 18\).

Step 8: Simplify the Expression

Combine like terms (terms with the same variable raised to the same power): \(x^2 + (9x + 2x) + 18 = x^2 + 11x + 18\).

Key Concepts

FOIL Method

The FOIL method is a quick way to multiply two binomials. The acronym FOIL stands for First, Outer, Inner, Last, highlighting the pairs of terms to multiply together. This method is commonly used in algebra for its simplicity and efficiency.

With the problem (x + 2)(x + 9) , we identify each part of the FOIL:

• First: Multiply the first terms of each binomial, which are both (x), giving (x^2).
• Outer: Multiply the outermost terms, (x) from the first binomial and (9) from the second, resulting in (9x).
• Inner: Multiply the inner pair of terms, (2) from the first binomial and (x) from the second, yielding (2x).
• Last: Multiply the last terms, (2) from both binomials, which gives (18).

After multiplying, the next step is to add each of these products together, resulting in the expanded expression (x^2 + 9x + 2x + 18).

Distribution in Algebra

Distribution in Algebra involves multiplying each term in one set of brackets by each term in another set of brackets. It's a fundamental technique that is often used to simplify expressions and solve equations efficiently.

In this example with (x + 2)(x + 9), each term of the first binomial is distributed across the second binomial:

• Distribute (x): Multiply (x) by each term in the second binomial ((x+9)), resulting in (x^2 + 9x).
• Distribute (2): Multiply (2) by each term in the same binomial ((x+9)), resulting in (2x + 18).

By distributing in this manner, we secure every combination of terms from both sets of brackets. After distribution, the terms are combined to give the final expression (x^2 + 11x + 18).

Expanding Expressions

Expanding expressions refers to the process of removing the parentheses in an algebraic expression and combining like terms. This is key when working with polynomials, as it helps in further simplification and evaluation of expressions.

When given the expression ((x+2)(x+9)), expanding means applying methods such as FOIL or distribution. Here, after applying such methods, we obtain (x^2 + 9x + 2x + 18).

Once an expression is expanded, the next crucial step is to simplify it by combining like terms. Similar terms share the same variables and powers. For this problem:

• Add (9x) and (2x) to form (11x).

This leads to the completely expanded and simplified expression (x^2 + 11x + 18), providing a clear polynomial expression with no parentheses and consolidated terms.