Problem 15 - Algebra A Combined Function

Question

Evaluate each polynomial when (a) \(x=0\) and \((b) x=-1 .\) See Examples 4 and 5. $$ x^{2}-5 x-2 $$

Step-by-Step Solution

Answer

When \(x = 0\), the polynomial evaluates to -2; when \(x = -1\), it evaluates to 4.

Step 1: Substitute and Evaluate for x=0

Replace every occurrence of \(x\) in the polynomial \(x^2 - 5x - 2\) with 0. This gives us \(0^2 - 5(0) - 2\). Calculating this expression: \(0 - 0 - 2 = -2\). So, when \(x = 0\), the polynomial evaluates to -2.

Step 2: Substitute and Evaluate for x=-1

Replace every occurrence of \(x\) in the polynomial \(x^2 - 5x - 2\) with -1. This gives us \((-1)^2 - 5(-1) - 2\). Calculating this expression: \(1 + 5 - 2 = 4\). So, when \(x = -1\), the polynomial evaluates to 4.

Key Concepts

Substitute Values

When working with polynomials, one crucial step involves substituting given values into the expression. This means replacing the variable in the polynomial with specific numbers provided in the question. For the polynomial \(x^2 - 5x - 2\), if you're asked to substitute \(x = 0\), you replace every \(x\) with 0. This turns the expression into \(0^2 - 5(0) - 2\).

• First, you perform the operations inside the expression, such as exponentiation and multiplication.
• Next, you simplify it by continuing with subtraction or addition.

This concept of substitution is essential not only in polynomial evaluation but also in other areas of algebra. By replacing variables with numbers, you get a clearer understanding of how the polynomial behaves with different inputs.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operations. The expression \(x^2 - 5x - 2\) is a polynomial, which is a specific type of algebraic expression.

• It has three terms: \(x^2\), \(-5x\), and \(-2\).
• "\(x^2\)" is called a quadratic term because it has an exponent of 2.
• "\(-5x\)" is linear because the variable is to the power of 1.
• The "constant term" is \(-2\), as it doesn’t involve a variable.

These components are joined by addition or subtraction to form the entire expression. Understanding how to read and interpret each part of an algebraic expression will help you when you simplify and evaluate it.

Simplifying Polynomials

After substituting values into a polynomial, the next step is simplifying the equation to find the result. Simplifying means performing the operations in a predetermined order (known as "order of operations") until you reach a single number.

• First, handle any exponentiations, such as squaring.
• Next, do any multiplications or divisions.
• Finally, perform any additions or subtractions.

In the given polynomial \(x^2 - 5x - 2\), when \(x = 0\), the polynomial simplifies directly to \(-2\). For \(x = -1\), the steps are: calculate \((-1)^2 = 1\), then \(-5 \times -1 = 5\), finally add these results with \(-2\) yielding \(1 + 5 - 2 = 4\).
Simplification allows you to see how the polynomial responds to different values and is fundamental in algebra.