91Ó°ÊÓ

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. These symbols represent numbers without specifying their exact values. In algebra, we use letters (like x or y) to represent unknown values or variables, which allows us to create general mathematical statements called equations or functions. Understanding algebra is essential because it's the foundation for many other areas of mathematics, including polynomial functions and calculus. Here, we deal with the algebraic function:

• Identifying the given function
• Substituting a value into the function
• Simplifying the expression
• Finding the final value

We see these steps in action as we substitute the value -1 into the function p(x) = -x^3 - x^2 + 4x.

A polynomial function is a function that involves only non-negative integer powers of x. The general form is: \(a_nx^n + a_{n-1}x^{n-1} + \text{...} + a_1x + a_0\) where \(a_n, a_{n-1}, ... , a_1, a_0\) are constants and \(n\) is a non-negative integer. In this exercise, we work with a polynomial function: \(p(x) = -x^3 - x^2 + 4x\) This function is a polynomial because it is composed of terms that are powers of x with coefficients. Polynomial functions have various applications in real life, including physics, engineering, and economics. Key points to remember about polynomial functions are:

• They can have multiple terms.
• The highest power of x determines the degree of the polynomial.
• Polynomials are continuous and smooth functions.

In our specific example, evaluating the polynomial at x=-1 was straightforward because it involved simple arithmetic of given coefficients and powers.

The substitution method is a technique used to evaluate the value of a function for a given input. Here’s how it works:

• Identify the function to be evaluated.
• Substitute the specific value into the function wherever the variable appears.
• Simplify the resulting expression to find the final value.

We used the substitution method here when we replaced \(x\) with \(-1\) in the function \(p(x)\). This gave us:\(p(-1) = -(-1)^3 - (-1)^2 + 4(-1)\)We then simplified this to find:

• \((-1)^3 = -1\)
• \((-1)^2 = 1\)
• \(4(-1) = -4\)

Thus, simplifying further, we got the result \(p(-1) = 1 - 1 - 4 = -4\). The substitution method is immensely useful in various algebraic problems, making it a fundamental skill for students to master.