Problem 6 - Algebra 2 and Trigonometry

Question

In \(3-12,\) write the sum or difference of the given polynomials in simplest form. $$ \left(-x^{2}+5 x+8\right)+\left(x^{2}-2 x-8\right) $$

Step-by-Step Solution

Answer

The simplified form of the polynomial sum is \(3x\).

Step 1: Understand the Problem

We are given two polynomials, \(-x^2 + 5x + 8\) and \(x^2 - 2x - 8\), which we need to add together. Our goal is to combine the like terms and express the resulting polynomial in its simplest form.

Step 2: Identify Like Terms

Like terms are those that have the same variable raised to the same power. In this expression, the like terms are \(-x^2\) and \(x^2\), \(5x\) and \(-2x\), and \(8\) and \(-8\).

Step 3: Combine Like Terms

Combine the like terms:- For the \(x^2\) terms: \(-x^2 + x^2 = 0\)- For the \(x\) terms: \(5x - 2x = 3x\)- For the constant terms: \(8 - 8 = 0\).

Step 4: Write the Simplified Polynomial

After combining all the like terms, the polynomial simplifies to \(3x\). The terms \(-x^2 + x^2\) and \(8 - 8\) both equal zero, so they are eliminated from the expression.

Key Concepts

Combining Like Terms

When dealing with polynomials, one of the important skills to master is combining like terms. But what exactly are like terms?

• Like terms are parts of an expression that have the same variable raised to the same power. For instance, in the expression given, terms \(-x^2\) and \(x^2\) are like terms because they both involve \(x\) squared.
• Similarly, \(5x\) and \(-2x\) are like terms since they both have the variable \(x\) raised to the first power.
• Numbers without variables, known as constant terms like 8 and -8, are also considered like terms.

To combine like terms, add or subtract their coefficients (the numbers in front of the variables). This process helps to simplify the polynomial by reducing it to fewer terms.

Simplified Form

Once you've combined all like terms, your objective is to express the polynomial in its simplest form. Let's break down what 'simplified form' actually means.

• In math, a simplified form of a polynomial is an expression with no like terms left uncombined and no unnecessary terms.
• In the given problem, after combining the like terms, the polynomial simplified to \(3x\), which means all operations were completed, and no further simplification was possible.
• This also means all zero-sum pairs, like \(-x^2 + x^2 = 0\) and \(8 - 8 = 0\), were effectively eliminated from the expression.

A simplified polynomial is easier to understand and more straightforward to work with, especially when performing further arithmetic operations or solving equations.

Polynomials

To delve into polynomial operations, it's essential first to grasp what a polynomial is.

• A polynomial is a mathematical expression involving a sum of powers of one or more variables multiplied by coefficients.
• Each of these sums is called a term, and the value of the polynomial is dependent on the value of the variable(s).
• For example, in the expression \(-x^2 + 5x + 8\), each part \(-x^2\), \(5x\), and \(8\) are individual terms of the polynomial.

Understanding how to add or subtract polynomials by combining like terms is crucial because it consolidates multiple terms into a more streamlined expression. Whether you're preparing for further algebra studies or just brushing up on basic math skills, recognizing polynomial forms is an invaluable skill in both.