Problem 52 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 2 and \(3 .\) \(3 s^{2}-2 s+8\) for a. \(\quad s=1\) b. \(s=0\)

Step-by-Step Solution

Answer

For \(s = 1\), the result is 9. For \(s = 0\), the result is 8.

Step 1: Substitute in the Expression for s=1

First, substitute \(s = 1\) into the expression \(3s^2 - 2s + 8\). The expression becomes \(3(1)^2 - 2(1) + 8\).

Step 2: Simplify the Expression for s=1

Now simplify each term: \(3(1)^2 = 3\), \(-2(1) = -2\), and the constant \(+8\) remains the same. Thus, the expression becomes \(3 - 2 + 8\).

Step 3: Calculate the Result for s=1

Add the simplified values: \(3 - 2 = 1\), and then \(1 + 8 = 9\). So, the result for \(s = 1\) is \(9\).

Step 4: Substitute in the Expression for s=0

Substitute \(s = 0\) into the expression \(3s^2 - 2s + 8\). The expression becomes \(3(0)^2 - 2(0) + 8\).

Step 5: Simplify the Expression for s=0

For \(s=0\), all terms involving \(s\) will become zero: \(3(0)^2 = 0\), \(-2(0) = 0\), and the constant \(+8\) remains the same. So, the expression simplifies to \(0 + 0 + 8\).

Step 6: Calculate the Result for s=0

Add the simplified values: \(0 + 0 = 0\), and \(0 + 8 = 8\). Therefore, the result for \(s = 0\) is \(8\).

Key Concepts

Substitution

Substitution is a key mathematical process where you replace a variable in an expression with a specific value. In the given exercise, we are tasked with evaluating the expression \(3s^2 - 2s + 8\) by substituting the variable \(s\) with different numerical values. This involves taking the expressions like \(s=1\) or \(s=0\), and placing these values directly into the expression in place of the variable \(s\).

This process helps in determining the outcome of an expression under specific conditions. In this case:

• For \(s = 1\), the expression becomes \(3(1)^2 - 2(1) + 8\).
• For \(s = 0\), it becomes \(3(0)^2 - 2(0) + 8\).

Substitution is simple yet powerful, allowing you to test expressions across different scenarios, which is essential for understanding how equations behave with different values.

Simplification

Simplification means reducing a mathematical expression to its simplest form. This involves performing operations and combining like terms in a logical sequence. After substituting a value for \(s\) in our polynomial, we need to simplify it to find the result.

For example, once we substitute \(s = 1\), the expression \(3(1)^2 - 2(1) + 8\) simplifies by calculating each term:

• Calculate \(3(1)^2\), which is \(3\).
• Calculate \(-2(1)\), which is \(-2\).
• The constant term \(+8\) stays the same.

Combine these calculated values: \(3 - 2 + 8\), resulting in the simplified value of \(9\). This approach is repeated for each substitution to make sure we accurately arrive at the simplest form.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, or division. Algebraic expressions such as \(3s^2 - 2s + 8\) can model real-world problems and abstract thinking.

Understanding algebraic expressions involves knowing how to manipulate them using substitution and simplification. Terms like \(3s^2\) and \(-2s\) use the variable \(s\), which makes them crucial when doing operations like substitution. The expression's structure helps convey relationships and interactions between numbers and variables.

Evaluating these expressions with certain values or conditions provides insights into how the variables impact the overall expression, enhancing problem-solving and analytical skills.

Polynomial

A polynomial is a specific type of algebraic expression formed by sums and differences of terms, each consisting of a coefficient, a variable raised to a non-negative integer exponent, and operations. In this exercise, \(3s^2 - 2s + 8\) is a polynomial with:

• Three terms: \(3s^2\), \(-2s\), and \(+8\).
• A degree of 2, as the highest exponent of the variable \(s\) is 2.

Polynomials are foundational in algebra, serving as the basis for further topics like factoring, polynomial division, and calculus.

Working with polynomials involves using operations such as addition, subtraction, and multiplication. It also includes understanding how to substitute variables and simplify the resulting expression without changing its equality. Mastery of polynomials is essential for advanced mathematical studies and practical applications.

Constant Term

The constant term in an algebraic expression or polynomial is a value that remains unchanged, regardless of any substitutions made to the other variables. It represents the quantity independent of the variable portion. In our expression \(3s^2 - 2s + 8\), the constant term is \(+8\).

This term is fixed and doesn't involve the variable \(s\), so in any substitution, this part of the expression stays the same. It is significant when calculating the result of an expression with a variable.

For example:

• When \(s = 1\), and you simplify, the constant \(+8\) remains as it is added to the other terms.
• Similarly, for \(s = 0\), the \(8\) remains unaffected.

Understanding the role of constant terms can help in grasping the behavior of expressions and equations. They provide stability to expressions as you manipulate other variable components.