Problem 43 - Elementary and Intermediate Algebra

Question

Use vertical form to add the polynomials. $$ \begin{array}{l} {6 a^{2}+7 a+9} \\ {-9 a^{2}} \quad {-2} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(-3a^2 + 7a + 7\).

Step 1: Align the Polynomials

Write the polynomials such that corresponding terms are aligned vertically. We have: \( 6a^2 + 7a + 9 \) and \( -9a^2 - 2 \). Place these in vertical form matching terms: \[\begin{array}{r} 6a^{2} + 7a + 9 \-9a^{2} + 0a - 2 \\hline\end{array}\] Notice that \(-9a^2\) is directly under \(6a^2\) and \(-2\) is under \(9\). We add a \(0a\) term to align with the \(7a\) term.

Step 2: Add the Coefficients of Like Terms

Starting from the highest degree terms, add the coefficients together for each aligned column: 1. Add the \(a^2\) terms: \(6 + (-9) = -3a^2\) 2. Add the \(a\) terms: \(7 + 0 = 7a\) 3. Add the constant terms: \(9 + (-2) = 7\)

Step 3: Write the Resulting Polynomial

Combine the results from the addition of coefficients to write the new polynomial: \(-3a^2 + 7a + 7\).

Key Concepts

Vertical Form

Polynomials can be added more easily when they are arranged in a vertical form. This method is similar to how you might add two multi-digit numbers.
The main idea is to place like terms directly above each other, which makes adding their coefficients straightforward.
This approach reduces mistakes commonly made in horizontal addition, like forgetting a term or misaligning others.

• First, identify the polynomial terms and stack them vertically.
• Align terms by their degree, starting with the highest degree to the lowest.
• Fill in any missing terms with a "0" to ensure every column is aligned correctly.

Like Terms

Like terms are terms within an algebraic expression that have the exact same variables raised to the same power.
Identifying them is crucial for combining them through addition or subtraction.
For example, in the expressions given, the terms containing \(a^2\), \(a\), and the constant terms are like terms.

• For the expression \(6a^2 + 7a + 9\), \(6a^2\) and \(-9a^2\) are like terms.
• Terms like \(7a\) and \(0a\) are aligned to be combined.
• Constant numbers such as \(9\) and \(-2\) are also like terms because they're both constants.

Understanding and identifying like terms ensures you are combining similar elements, which simplifies the whole operation.

Coefficient Addition

Adding coefficients is a simple yet crucial step.
A coefficient is the numerical part of a term that multiplies the variable.
When adding polynomials, you specifically focus on adding the coefficients of like terms together.

• The \(a^2\) terms' coefficients: \(6 + (-9) = -3\).
• The \(a\) term involves adding \(7\) and \(0\) (since there is no second \(a\)), resulting in \(7a\).
• Finally, for the constant terms, add \(9\) and \(-2\) to get \(7\).

Each column in the vertical form corresponds to a specific term in the polynomial, and adding these coefficients gives you the result of that term.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations.
In the case of polynomials, these expressions consist of one or more terms.
Understanding algebraic expressions is fundamental to manipulating them through operations like addition or subtraction.

• They can include constants, coefficients, and variables, such as \(6a^2\), \(7a\), and \(9\).
• When arranged correctly, the forms of these expressions guide the methods used for operations like addition.
• Learning to handle these expressions prepares students for more complex algebraic manipulations.

By practicing operations on algebraic expressions, you'll gain skills for handling more advanced mathematical problems.