Problem 51 - Elementary and Intermediate Algebra

Question

Evaluate each expression. See Example 2 and \(3 .\) \(4 t^{2}+2 t-8\) for a. \(t=-1\) b. \(t=0\)

Step-by-Step Solution

Answer

a. \(-6\); b. \(-8\)

Step 1: Substitute the value of t in Expression a

First, we need to substitute \( t = -1 \) into the expression \( 4t^2 + 2t - 8 \).Substitute: \( 4(-1)^2 + 2(-1) - 8 \).

Step 2: Simplify the expression for a

Calculate \((-1)^2\) which gives \(1\), so \(4(-1)^2 = 4 \ imes 1 = 4\).Next, multiply: \(2(-1) = -2\).Subtract: \( 4 - 2 - 8 \).

Step 3: Evaluate the resulting expression for a

Combine the terms: \( 4 - 2 = 2 \).Subtract: \( 2 - 8 = -6 \).So, when \( t = -1 \), the expression evaluates to \(-6\).

Step 4: Substitute the value of t in Expression b

Now we substitute \( t = 0 \) into the expression \( 4t^2 + 2t - 8 \).Substitute: \( 4(0)^2 + 2(0) - 8 \).

Step 5: Simplify the expression for b

Calculate \((0)^2\) which gives \(0\), so \(4(0)^2 = 4 \ imes 0 = 0\).Next, multiply: \(2(0) = 0\).Subtract: \( 0 + 0 - 8 \).

Step 6: Evaluate the resulting expression for b

Combine the terms: \( 0 + 0 = 0 \).Subtract: \( 0 - 8 = -8 \).So, when \( t = 0 \), the expression evaluates to \(-8\).

Key Concepts

Polynomials

Polynomials are mathematical expressions that consist of variables and coefficients, structured using addition, subtraction, and multiplication. They are a cornerstone of algebra and enable us to express a wide array of mathematical relationships. A polynomial can have different terms, each being a product of a number (called a coefficient) and a variable raised to a power. For example, in the expression \(4t^2 + 2t - 8\), there are three terms:

• \(4t^2\): The term has a coefficient of \(4\) and a variable \(t\) raised to the power of \(2\).
• \(2t\): This term has a coefficient of \(2\) and a variable \(t\) raised to the power of \(1\).
• \(-8\): This is the constant term, which does not involve the variable.

Polynomials are often categorized by their degree, which is the highest power of the variable in the expression. For instance, \(4t^2 + 2t - 8\) is a quadratic polynomial because the highest degree is \(2\). Understanding polynomials helps in simplifying and solving equations through various algebraic techniques.

Evaluating Expressions

Evaluating expressions is a process that involves finding the value of an algebraic expression by substituting the variable(s) with specific number(s). This enables us to understand the behavior of the expression under different conditions. When evaluating, we need to carefully follow the order of operations, often abbreviated as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we simplify the expression correctly.Let's break it down:

• Substitution: Replace the variable in the expression with the given number.
• Operations: Follow the order of operations to simplify the expression step-by-step.
• Result: Once simplified, you'll obtain the final numerical value.

For the expression \(4t^2 + 2t - 8\), we solve for specific values of \(t\). By substituting \(t = -1\) and performing the arithmetic, we gravitate towards understanding how changes in \(t\) impact the polynomial's outcome.

Substitution Method

The substitution method is a useful technique in algebra for evaluating expressions by replacing variables with specific numbers. This method is straightforward and crucial for simplifying polynomial expressions step-by-step. Here’s a closer look at how this method works:1. **Identify the variable and its value:** When the exercise specifies a value for the variable, like \(t = -1\) or \(t = 0\), note it down.2. **Replace the variable:** Substitute the indicated values into the variables of the expression. For example, substitute \(t = -1\) in \(4t^2 + 2t - 8\), yielding \(4(-1)^2 + 2(-1) - 8\).3. **Simplify step-by-step:** Carry out the arithmetic operations, following the order of operations, until you arrive at a single numerical value. This involves evaluating powers first, then multiplication, and finally addition or subtraction. For instance, calculating \(4(-1)^2\) as \(4 \, \times \, 1\) which equals \(4\), then proceeding with the remaining calculations.By practicing this substitution method, you can tackle the challenge of evaluating both simple and complex polynomials with confidence and accuracy. It aids in visualizing how the polynomial transforms based on variable changes.