Problem 3 - College Algebra

Question

You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?

Step-by-Step Solution

Answer

Distribute, apply the distributive property, combine like terms, and simplify.

Step 1: Distribute Each Term

Begin by distributing each term in the first polynomial to every term in the second polynomial. This means taking each term separately from the first polynomial and performing multiplication across all terms present in the second polynomial.

Step 2: Apply the Distributive Property

For each multiplication identified in Step 1, use the distributive property to ensure that each term in the first polynomial multiplies with each term in the second polynomial. This involves applying the formula: \( a(b+c) = ab + ac \). Ensure that each term from one polynomial is multiplied by each term of the other polynomial.

Step 3: Combine Like Terms

Once all terms are multiplied, you will have an expanded expression. Look through the resulting terms and combine any like terms. Like terms will have the exact same variable components. Add or subtract the coefficients of these terms to simplify the polynomial.

Step 4: Simplify the Expression

Finally, check the entire expression for any mistakes and ensure all like terms have been combined. The result should be the expanded polynomial expression with the simplest terms.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in mathematics that helps us simplify multiplication across sums. When you're multiplying polynomials, this property is a key tool.
Imagine you have two polynomials: one is \(a + b\), and the other is \(c + d\). You can use the distributive property to expand this product. In other words, you’ll distribute each term in the first polynomial across each term in the second polynomial.
For example, if you take the first term \(a\), you will multiply it with both \(c\) and \(d\). Similarly, the second term \(b\) will also multiply with both \(c\) and \(d\). This results in:

• \(a \, \times \, c\)
• \(a \, \times \, d\)
• \(b \, \times \, c\)
• \(b \, \times \, d\)

This approach ensures you haven't missed multiplying any terms, fulfilling the distributive property: \(a(b+c) = ab + ac\). Keep in mind: always ensure each term from one polynomial interacts with every term in the other!

Combining Like Terms

After applying the distributive property, you often end up with several terms. Some of these terms will be the same or similar with slight differences in their coefficients. That's where we use the technique called "combining like terms."
Like terms are those that have the same variable raised to the same power. For instance, \(3x^2\) and \(5x^2\) are like terms because they both contain \(x^2\). To combine them, simply add or subtract their coefficients:

• \(3x^2 + 5x^2 = 8x^2\)

This process reduces the number of terms you're working with, making your polynomial easier to understand and work with. Look out for similar variables and powers, and handle them systematically to simplify the expression effectively.

Simplifying Expressions

Once you’ve successfully combined like terms, the last step is to simplify the expression. Simplifying means reducing to the smallest and most straightforward form possible.
This step ensures your answer is clean and concise.
Here's a small checklist to help simplify effectively:

• Make sure all like terms are combined.
• Check for any possible factorization.
• Ensure there are no mistakes in the sign (positive or negative).
• Write the expression in descending order based on the power of terms, if applicable.

Simplifying not only aids in clarity but also is essential for verifying your answer’s correctness. Double-check your expression, ensuring everything adds up right and reads smoothly.