Problem 8 - Elementary and Intermediate Algebra

Question

What is the result of the addition in the \(x\) -column? $$ \begin{array}{l} {4 x^{2}+x-12} \\ {5 x^{2}-8 x+23} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The result is \(9x^2 - 7x + 11\).

Step 1: Lay out the expression.

We have two expressions to add in the \(x\)-column: \(4x^2 + x - 12\) and \(5x^2 - 8x + 23\).

Step 2: Identify like terms.

The like terms in the given expressions are \(4x^2\) and \(5x^2\), \(x\) and \(-8x\), and \(-12\) and \(+23\) respectively.

Step 3: Add the like terms for \(x^2\).

Add the coefficients of the \(x^2\) terms: \(4x^2 + 5x^2 = 9x^2\).

Step 4: Add the like terms for \(x\).

Combine the coefficients of the \(x\) terms: \(1x - 8x = -7x\).

Step 5: Add the constant terms.

Add the constants: \(-12 + 23 = 11\).

Step 6: Write the resulting expression.

Combine all the results from steps 3, 4, and 5: The resultant expression is \(9x^2 - 7x + 11\).

Key Concepts

Understanding Like Terms

When working with polynomial addition, it is crucial to identify and group like terms. Like terms are the parts of an algebraic expression that have the same variables raised to the same power. For example, in the polynomial expression \(4x^2 + x - 12\), the term \(4x^2\) shares the same variable and exponent as \(5x^2\) from the second expression \(5x^2 - 8x + 23\).

This means they are like terms and can be added together to simplify the expression. The same logic applies to the linear terms \(x\) and \(-8x\), as well as the constant terms \(-12\) and \(23\).

When performing polynomial addition, always follow these steps:

• Identify terms that have the same variable and exponent.
• Group these similar or 'like' terms together.
• Then proceed to add or subtract them separately from other groups of terms.

This method ensures clarity in simplifying any polynomial expression.

Importance of Coefficients

In algebraic expressions, coefficients are the numerical parts of the terms. They tell us how many times a term will be counted. For instance, in \(4x^2\), the coefficient is 4, indicating that \(x^2\) is taken 4 times.

Understanding coefficients is vital, as it helps to correctly and efficiently perform operations like addition. When combining terms, you only add or subtract their coefficients if the terms are like terms.

• For \( x^2 \) terms: Add \(4x^2\) and \(5x^2\) by adding the coefficients: 4 + 5 = 9, which gives \(9x^2\).
• For linear \(x\) terms: Combine \(x\) with \(-8x\) by subtracting their coefficients: 1 - 8 = -7, resulting in \(-7x\).

Without understanding the role of coefficients, the addition of polynomials could become confusing and error-prone.

Algebraic Expressions Explained

An algebraic expression is a mathematical phrase combining numbers, variables, and operators such as addition and subtraction. Expressions include terms which can be constants, variables, or products of variables with numerical coefficients.

In the example from the exercise, two algebraic expressions are being added. Each expression consists of three parts: quadratic terms \((x^2)\), linear terms \((x)\), and constant terms. The ultimate goal is to combine the expressions into one simplified polynomial.

Here’s how it’s done:

• Break down each individual expression into its constituent parts.
• Identify the like terms across the expressions.
• Add coefficients of like terms to yield a consolidated algebraic expression.

Ultimately, understanding algebraic expressions and their components helps in organizing and simplifying polynomials effectively, a cornerstone of algebra.