Problem 42 - Beginning and Intermediate Algebra

Question

Can FOIL be used to expand \((x+9)^{2}\) ? Explain your answer.

Step-by-Step Solution

Answer

Yes, FOIL can be used to expand \((x+9)^{2}\), as it can be rewritten as the multiplication of two binomials \((x+9)(x+9)\). Applying the FOIL method, we get \(x^2+18x+81\).

Step 1: Rewrite the expression

Let's rewrite \((x+9)^{2}\) as the multiplication of two binomials. Since \((x+9)^{2}\) is the square of \((x+9)\), we can rewrite it as \((x+9)(x+9)\).

Step 2: Apply the FOIL method

Now, we will apply the FOIL method to the expression \((x+9)(x+9)\). FOIL stands for First, Outer, Inner, Last: 1. First, multiply the first terms of both binomials: \(x \cdot x = x^2\). 2. Outer, multiply the outer terms of both binomials: \(x \cdot 9 = 9x\). 3. Inner, multiply the inner terms of both binomials: \(9 \cdot x = 9x\). 4. Last, multiply the last terms of both binomials: \(9 \cdot 9 = 81\).

Step 3: Combine the terms

Now, we will combine the results of all four multiplications, adding like terms where necessary: \(x^2+ 9x+ 9x+ 81\) We will now simplify the expression by combining the similar terms: \(x^2+ 18x+ 81\) Since we were able to use the FOIL method to expand \((x+9)^{2}\), we can conclude that the answer is yes, FOIL can be used to expand \((x+9)^{2}\). The result is \(x^2+18x+81\).

Key Concepts

Binomial Expansion

Binomial expansion refers to the method of multiplying out and simplifying a binomial expression raised to a power. It's particularly useful when dealing with expressions like \((a+b)^{n}\). In this method, every term in the first binomial is multiplied by every term in the second binomial. For example, with \((x+9)^{2}\), we can expand it using the FOIL method. This stands for First, Outer, Inner, Last, and helps ensure all pairs of terms are multiplied correctly.

• First Terms: Multiply the first terms of each binomial.
• Outer Terms: Multiply the first term of the first binomial with the second term of the second binomial.
• Inner Terms: Multiply the second term of the first binomial with the first term of the second binomial.
• Last Terms: Multiply the last terms of both binomials.

This process allows you to expand and combine terms into a single polynomial, providing a clear understanding of how the terms within a binomial expression interact after expansion.

Polynomials

Polynomials are algebraic expressions that consist of terms made up of constants and variables, which are combined using addition, subtraction, multiplication, and non-negative integer exponents. They can have multiple terms; for example, a binomial is a polynomial with two terms. A polynomial like \(x^2 + 18x + 81\) results from binomial expansion, representing a second-degree polynomial.

• Terms: Parts of an expression separated by a plus or minus sign. For example, \(x^2\), \(18x\), and \(81\) are each terms.
• Degree of Polynomial: The highest exponent of the variable in the expression. In \(x^2 + 18x + 81\), the degree is 2 because of \(x^2\).
• Coefficients: Numbers in front of variable terms. Here, '18' is the coefficient of \(x\).

Polynomials form the backbone of algebra and calculus, providing a foundation for solving a wide range of mathematical problems.

Algebraic Expressions

Algebraic expressions are composed of variables, numbers, and mathematical operators. They form the basis of equations and functions in algebra. Understanding them is critical to solving mathematical problems involving variables. In \((x + 9)(x + 9)\), each part is defined:

• Variables: Symbols such as \(x\), which represent numbers.
• Constants: Numbers without variables, like 9 in the expression \(x+9\).
• Operators: Symbols indicating math operations, such as addition \(+\) or multiplication \(\cdot\).

These expressions facilitate forming equations and performing manipulations like expanding or simplifying. The FOIL method is a practical tool for handling binomial expressions, helping convert them into expanded polynomial forms by considering all paired combinations of algebraic terms.