Problem 22 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\left(3 x^{2}-4 x+5\right)+\left(-2 x^{2}+3 x-2\right)$$

Step-by-Step Solution

Answer

The sum of the expressions is \(x^2 - x + 3\).

Step 1: Identify like terms

Like terms in an algebraic expression are terms that have the same variable raised to the same power. In the given expression, the like terms are: \(3x^2\) and \(-2x^2\); \(-4x\) and \(3x\); \(5\) and \(-2\).

Step 2: Add/Subtract like terms

Now, we combine the like terms:\[3x^2 + (-2x^2) = (3 - 2)x^2 = x^2\]\[-4x + 3x = (-4 + 3)x = -x\]\[5 + (-2) = 5 - 2 = 3\]

Step 3: Write the final expression

Combine the simplified like terms into one expression:\[x^2 - x + 3\].

Key Concepts

Polynomial Addition

When working with algebra, polynomial addition is about combining two or more polynomials to form a single expression. A polynomial is a sum of terms where each term is a product of a coefficient and a variable raised to a non-negative integer power. In our exercise, we see two polynomials being added together: \( (3x^2 - 4x + 5) + (-2x^2 + 3x - 2) \). The key is to sequentially align each corresponding term based on the powers of the variables involved.The process:

• Ensure the polynomials are arranged clearly, often keeping them in descending order based on the power of the variable for convenience.
• Identify and group the like terms (terms with the same power of \( x \)).
• Perform the addition (or subtraction, if indicated) on each pair of like terms.

This simple process aids in seamlessly combining the polynomials into a singular expression.

Like Terms

Like terms are essential in simplifying polynomials. In algebra, a crucial rule is to only combine terms that have identical variables raised to the same powers. For instance, in the expression from our exercise \( 3x^2 - 4x + 5 \) and \( -2x^2 + 3x - 2 \), like terms are identified as follows:

• \( 3x^2 \) and \( -2x^2 \) are like terms because both have the variable \( x \) raised to the power of 2.
• \( -4x \) and \( 3x \) are like terms as they both have \( x \) to the power of 1.
• \( 5 \) and \( -2 \) are constants with no variables, making them like terms with only each other.

Recognizing like terms is crucial because it allows you to add or subtract them directly. This step simplifies the polynomial, making further operations more manageable.

Simplifying Expressions

The goal of many algebraic operations is to simplify expressions. Simplification means rewriting the expression in its simplest or most concise form. After identifying and combining like terms, as previously described, we simplify the polynomial.For the expression \(3x^2 - 4x + 5 \) summed with \(-2x^2 + 3x - 2 \), we simplify as follows:

• Combine the \( x^2 \) terms: \( 3x^2 + (-2x^2) = x^2 \).
• Combine the \( x \) terms: \( -4x + 3x = -x \).
• Combine the constant terms: \( 5 + (-2) = 3 \).

By doing these steps, we obtain the simplified polynomial: \( x^2 - x + 3 \). Simplifying makes the expression easier to interpret and use in further mathematical operations or solving equations.