Problem 22 - College Algebra with Modeling and Visualization

Question

Add the polynomials. $$\left(x^{3}-5 x^{2}+6\right)+\left(5 x^{2}+3 x+1\right)$$

Step-by-Step Solution

Answer

The sum is \( x^3 + 3x + 7 \).

Step 1: Write Down the Problem

We need to add the following polynomials: \( (x^3 - 5x^2 + 6) + (5x^2 + 3x + 1) \).

Step 2: Remove Parentheses

Remove the parentheses from each polynomial. This gives us: \( x^3 - 5x^2 + 6 + 5x^2 + 3x + 1 \).

Step 3: Group Like Terms

Group terms that have the same power of \( x \). So, we have: \( x^3 + (-5x^2 + 5x^2) + 3x + (6 + 1) \).

Step 4: Simplify Like Terms

Add the coefficients of the like terms together: \( x^3 + 0 \cdot x^2 + 3x + 7 \).

Step 5: Write the Final Answer

The sum of the polynomials is \( x^3 + 3x + 7 \).

Key Concepts

Understanding Polynomials

Polynomials are expressions that consist of variables and coefficients, combined using only addition, subtraction, and multiplication. The variables are often raised to whole number powers. Each term in a polynomial has a "degree," which is the sum of the exponents of the variables in that term. For example, in the polynomial \( x^3 - 5x^2 + 6 \), the term \( x^3 \) has a degree of 3, \( -5x^2 \) has a degree of 2, and the constant \( 6 \) is considered to have a degree of 0.

Polynomials can take many forms:

• Monomials: A single term, such as \( 2x \) or \( x^3 \).
• Binomials: Two terms, like \( x^2 - 4 \).
• Trinomials: Three terms, for instance, \( x^2 + 2x + 1 \).

Adding polynomials involves combining terms with the same degree, known as 'like terms,' to simplify the overall equation. This creates a new polynomial that is often more straightforward in its representation.

The Art of Combining Like Terms

Combining like terms is a crucial step in simplifying polynomials. Like terms are terms that have the same variable raised to the same power. For instance, in the polynomial from our exercise, the terms \(-5x^2\) and \(5x^2\) both have the variable \(x^2\).

To combine these terms, you simply add or subtract their coefficients (the numbers in front of the variables). So, in our example:

• \(-5x^2 + 5x^2 = 0\)

This cancellation occurs because adding a positive and negative number of the same magnitude results in zero.

This step reduces the complexity of the expression and is essential for achieving a simplified form of the polynomial. After combining like terms, you'll usually arrange the terms in order of descending powers, starting with the highest.

Basic Algebra and Polynomial Manipulation

Algebra is the branch of mathematics that deals with symbols and rules for manipulating those symbols. It is essential for solving equations, including those involving polynomials. Polynomials are often encountered in algebra as they are foundational to many algebraic operations. Understanding algebra makes it easier to perform operations on polynomials like addition, which we performed in the exercise.

Basic algebraic skills include:

• Understanding how to handle variables and constants.
• Performing arithmetic operations with variables, like addition and subtraction.
• Recognizing equivalent expressions and simplifying them.

In our exercise, we used these skills to transform the original polynomials by removing parentheses and combining like terms. Through these algebraic techniques, we successfully added the polynomials to arrive at the simplified answer \( x^3 + 3x + 7 \), illustrating the power and usefulness of algebra in manipulating and simplifying expressions.