Problem 24 - College Algebra with Modeling and Visualization

Question

Add the polynomials. $$\left(y^{5}+y\right)+\left(5-y+\frac{1}{3} y^{2}\right)$$

Step-by-Step Solution

Answer

The result is \(y^5 + \frac{1}{3}y^2 + 5\).

Step 1: Remove Parentheses

First, remove the parentheses in the polynomial expression: \((y^5 + y) + (5 - y + \frac{1}{3}y^2)\). This gives us: \(y^5 + y + 5 - y + \frac{1}{3}y^2\).

Step 2: Combine Like Terms

Now identify and combine the like terms from the expression. The terms are: \(y^5\), \(y - y = 0\), and \(\frac{1}{3}y^2\), along with the constant 5. So we'll combine these to get: \(y^5 + \frac{1}{3}y^2 + 5\).

Step 3: Simplify the Expression

Write down the simplified form of the polynomial by combining the results of the like terms: \(y^5 + \frac{1}{3}y^2 + 5\).

Key Concepts

Combining Like Terms

In algebra, especially when dealing with polynomial expressions, combining like terms is a fundamental skill. But what exactly does it mean to combine like terms?

**Like Terms Defined**
Like terms are terms that have the exact same variable(s) raised to the same power. This means both the variable part and the exponent must match. For instance, in the expression \(3x^2 + 5x^2 - 2x\), the terms \(3x^2\) and \(5x^2\) are like terms because they both have the variable \(x\) raised to the power of two. However, \(-2x\) is not a like term with \(3x^2\) because it doesn't share exactly the same exponent.

**Combining the Terms**
Once you've identified like terms, you can combine them by adding or subtracting their coefficients. In the provided example, the expression \( (y^5 + y) + (5 - y + \frac{1}{3}y^2) \), identifies like terms: \( y \) and \( -y \). These cancel each other out, simplifying the term to zero. This step is crucial for making expressions easier to work with in future calculations.

Simplifying Expressions

Simplifying expressions is the process of making an algebraic expression as neat and concise as possible. This usually involves combining like terms, but can also include removing unnecessary brackets and simplifying any fractions.

**Steps in Simplification**

• **Remove Parentheses**: Start by expanding the expression if necessary. This means taking the expression out of any parentheses.
• **Combine Like Terms**: Gather all the like terms you can find and combine them by adding or subtracting coefficients.
• **Organize the Expression**: Write down the simplified terms in order, often from the highest to the lowest degree of variables.

In the exercise, simplification resulted in \( y^5 + \frac{1}{3}y^2 + 5 \), with terms arranged neatly according to their degrees.

Polynomial Expressions

Polynomial expressions are algebraic expressions that include sums of variables raised to whole number exponents. Each summand is called a "term," and it might comprise constants, variables, or both.

**Understanding Polynomials**

• **Terms**: Each term includes a coefficient and the variable raised to an exponent. In \(3x^2 + 2x + 1\), we see three terms.
• **Degree**: The degree of a polynomial is the highest exponent of its variables. For \(y^5 + \frac{1}{3}y^2 + 5\), the highest degree is 5.
• **Standard Form**: Polynomials are often written starting from the highest degree to the lowest.

Polynomials can be added, subtracted, multiplied, and divided, with various methods to simplify them. The given problem involved polynomial addition and simplification, crucial skills for more advanced algebra.