Problem 14 - Algebra for College Students

Question

Simplify the algebraic expressions in Problems \(1-14\) by combining similar terms. $$8 x y^{2}-5 x^{2} y+2 x y^{2}+7 x^{2} y$$

Step-by-Step Solution

Answer

The simplified expression is \(10xy^2 + 2x^2y\)."

Step 1: Identify Similar Terms

We have the expression \(8xy^2 - 5x^2y + 2xy^2 + 7x^2y\). Observe that terms with the same base and the same powers of variables are similar. Here, \(8xy^2\) and \(2xy^2\) are similar terms, and \(-5x^2y\) and \(7x^2y\) are similar terms.

Step 2: Combine Similar Terms - First Set

Combine the similar terms \(8xy^2\) and \(2xy^2\) by adding their coefficients. So, \(8 + 2 = 10\), which gives us \(10xy^2\).

Step 3: Combine Similar Terms - Second Set

Next, combine \(-5x^2y\) and \(7x^2y\) by adding their coefficients: \(-5 + 7 = 2\). This results in \(2x^2y\).

Step 4: Write the Simplified Expression

Put the combined terms from the previous steps into a single expression. We have \(10xy^2 + 2x^2y\).

Key Concepts

Combining Like Terms

In algebra, combining like terms is a process used to simplify expressions. This involves grouping terms that have the exact same combination of variables raised to the same powers. For instance, if you have variables combined like this: \(8xy^2 - 5x^2y + 2xy^2 + 7x^2y\), identifying similar terms is the first step towards simplification.

Similar terms have the same variables with the same powers, which means they look the same except for their coefficients. In our case:

• \(8xy^2\) and \(2xy^2\) are like terms because they both have the variable combination \(xy^2\).
• \(-5x^2y\) and \(7x^2y\) are also like terms with the variable combination \(x^2y\).

By recognizing these similar terms, you lay the foundation for further simplification of the expression.

Coefficient Addition

Once you've identified like terms, the next step is coefficient addition. The coefficient is the numerical part of a term. To combine like terms, you add their coefficients and retain the variable part. This method is crucial for simplifying expressions accurately.

For the terms \(8xy^2\) and \(2xy^2\), you add the coefficients 8 and 2. Similarly, for \(-5x^2y\) and \(7x^2y\), the coefficients -5 and 7 are added.
Let's see this in action:

• \(8xy^2 + 2xy^2 = (8 + 2)xy^2 = 10xy^2\)
• \(-5x^2y + 7x^2y = (-5 + 7)x^2y = 2x^2y\)

Coefficient addition changes the expression from a jumble of terms to something more streamlined, making it much easier to handle.

Polynomial Simplification

Polynomial simplification is the end goal when combining and adding like terms. Once like terms are combined using coefficient addition, what remains should be a simpler version of the polynomial expression.

After performing coefficient addition with our terms, we arrive at the expression \(10xy^2 + 2x^2y\). This is the simplified form of our original polynomial. Simplifying polynomials helps in various mathematical processes like solving equations or evaluating expressions at specific values, since it gives us a much cleaner form to work with.

• Simplified polynomials are easier to understand and compute.
• They reveal the structure of an expression, helping in further algebraic manipulations.

By following this structured approach, you can reduce complex expressions to their simplest forms, facilitating deeper mathematical insight and ease of computation.