Problem 104 - Algebra and Trigonometry

Question

In Exercises 103–106, determine whether each statement makes sense or does not make sense, and explain your reasoning. I used the FOIL. method to find the product of \(x+5\) and \(x^{2}+2 x+1\)

Step-by-Step Solution

Answer

Yes, the student's statement makes sense. Applying the FOIL method to find the product of \(x+5\) and \(x^2+2x+1\), though normally used for binomials, can be extended for this purpose. The resulting product will be: \(x^3+7x^2+11x+5\).

Step 1: Align the Polynomials for multiplication

The first step in multiplying polynomials is to align them so you can clearly see the multiplication process. This would look like this: \((x+5) * (x^2+2x+1)\).

Step 2: Apply FOIL method to compute the product

Start by multiplying each term in the first polynomial \(x+5\) by each term in the second polynomial \(x^2+2x+1\). It's essentially an extended version of the FOIL method. You should get: \(x*(x^2+2x+1) + 5*(x^2+2x+1)\).

Step 3: Distribute and Simplify

Expand each term: \(x^3+2x^2+x + 5x^2+10x+5\). Then combine like terms to get the final answer: \(x^3+7x^2+11x+5\).

Key Concepts

Multiplying Polynomials

When you are multiplying polynomials, you are taking two or more algebraic expressions and performing multiplication on every term. Each part of the first polynomial must be multiplied with each part of the second polynomial. In practice, you can imagine this as foiling out an area where each term of one polynomial represents one dimension and the terms of the other polynomial represent the other dimension. Here's a simplified approach:

• Write down each polynomial clearly.
• Take the first term of the first polynomial and multiply it by each term of the second polynomial. Do this for each term of the first polynomial.
• Record each result.

This process is not only crucial for polynomial multiplication but also strengthens the understanding of higher-level algebraic concepts.

Distributive Property

The distributive property is a cornerstone of algebra and is used to simplify complex multiplication. It states that, for all real numbers a, b, and c, the equation a(b + c) equals ab + ac holds true. In terms of polynomials, this means that each term inside a bracket can be multiplied by the term outside the bracket. The simplicity of this property masks its importance in polynomial arithmetic, as it allows for the systematic expansion of expressions. When multiple terms are involved, the distributive property must be applied to each term separately.
For example, in the exercise provided, when you multiply 5 by each term in the second polynomial, you use the distributive property to get 5 times each term, which are then summed up to get part of the final answer.

Combining Like Terms

After using the distributive property to expand the polynomial, you will often find that you have several terms that are similar, known as 'like terms.' Combining like terms is the process of simplifying an expression so that all like terms are combined into a single term. Like terms have the same variable raised to the same power.
The key step is to identify and sum up the coefficients of like terms.

• Look for terms with the same variables raised to the same power.
• Add or subtract the coefficients of these terms.
• Write the simplified expression with each term appearing only once.

This makes the final equation cleaner and easier to interpret, which is crucial for further calculations or interpretations.

Polynomial Arithmetic

Polynomial arithmetic is much like the arithmetic with numbers that we're all familiar with. It involves adding, subtracting, multiplying, and even dividing polynomial expressions using a set of rules and properties like the distributive property. Mastery of polynomial arithmetic is essential for problem solving in algebra and calculus. Here's how to approach it:

• Understand the rules for different operations, such as how to add or multiply polynomials.
• Apply the appropriate method for the operation, ensuring to carefully align terms and use the distributive property when needed.
• Combine like terms where possible to simplify the final expression.
• Double-check your final result to ensure it makes sense in the context of the problem.

With practice, manipulating these expressions becomes intuitive, setting a strong foundation for more advanced mathematics.