Problem 12 - Algebra Form and Function

Question

Find the degree. $$ 2 s^{3}-s^{2}+1-s^{3}+2 s^{2}-s+3 s^{3} $$

Step-by-Step Solution

Answer

Answer: The degree of the given polynomial is 3.

Step 1: Combine like terms

We will first combine the like terms, i.e., the terms with the same power of \(s\). $$ (2s^3 - s^3 + 3s^3) + (- s^2 + 2s^2) + (- s) + 1 $$

Step 2: Simplify the polynomial

Now let's simplify the polynomial by adding the coefficients of the like terms. $$ 4s^3 + s^2 - s + 1 $$

Step 3: Identify the highest power of s

In the simplified polynomial, let's identify the term with the highest power of \(s\). We have 3 terms with powers of \(s\): - \(4s^3\) - \(s^2\) - \(-s\) The highest power of \(s\) in the polynomial is \(s^3\) in the term \(4s^3\).

Step 4: Determine the degree of the polynomial

The degree of the polynomial is the highest power of the variable \(s\) that appears in its terms. In this case, the highest power of \(s\) is 3 (from the term \(4s^3\)). Therefore, the degree of the polynomial is 3.

Key Concepts

Degree of a Polynomial

The degree of a polynomial is crucial for understanding its properties and behavior. Simply put, the degree of a polynomial is the highest power of the variable present in the expression. In our exercise, the polynomial expression is simplified to \(4s^3 + s^2 - s + 1\). The degree tells us much about how the polynomial looks graphically and impacts its critical features.

• To find the degree, look for the highest exponent in the polynomial's terms.
• In the expression \(4s^3 + s^2 - s + 1\), the term \(4s^3\) has the highest exponent value of 3.
• Thus, the degree of this polynomial is 3. Remember, the degree signifies how many solutions there could potentially be and can help predict the graph's shape.

Understanding the degree helps in solving and graphing polynomials. It's the key to unraveling the polynomial's full potential.

Combining Like Terms

Combining like terms is an essential step in simplifying polynomials. Like terms are terms with the same power of the variable. In our example, this means gathering together common powers of \(s\) to create a more straightforward expression.

• First, identify terms that have identical variable powers.
• In the exercise, the initial polynomial is \(2s^3 - s^2 + 1 - s^3 + 2s^2 - s + 3s^3\).
• We gather like terms as \((2s^3 - s^3 + 3s^3)\), \((-s^2 + 2s^2)\), \(-s\), and \(+1\).
• After combining, you'll have \(4s^3 + s^2 - s + 1\).

This process leads to a more manageable and clear polynomial, paving the way for further computations.

Simplifying Expressions

Simplifying expressions means reducing them to their simplest form. This allows for easier manipulation and understanding. After combining like terms, simplifying involves arithmetic operations like adding or subtracting to reach this simplest form.

• In the step-by-step solution, simplification is demonstrated by adding the coefficients of combined terms.
• The expression \((2s^3 - s^3 + 3s^3)\) simplifies to \(4s^3\), the expression \((-s^2 + 2s^2)\) simplifies to \(s^2\).
• Finally, you arrive at \(4s^3 + s^2 - s + 1\).

Simplifying expressions is necessary to solve equations efficiently and helps in recognizing patterns and relationships within the polynomial. It's like cleaning up a messy room so you can see everything clearly.