Problem 44 - Elementary and Intermediate Algebra

Question

Use vertical form to add the polynomials. $$ \begin{array}{l} {-2 c^{2}-3 c-5} \\ {14 c^{2}} \quad \quad {-1} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(12c^2 - 3c - 6\).

Step 1: Write Each Polynomial in Column Form

Align the polynomials vertically, matching like terms. The given polynomials to add are:- The first polynomial is \(-2c^2 - 3c - 5\).- The second polynomial is \(14c^2 - 1\).Since the second polynomial lacks a \(c\) term, write it as \(14c^2 + 0c - 1\) for clarity when aligning:\[\begin{array}{c|c|c} & -2c^2 & -3c & -5 \\hline+ & 14c^2 & 0c & -1 \\hline\end{array}\]

Step 2: Add the \(c^2\) terms

Add the \(c^2\) terms: \(-2c^2 + 14c^2 = 12c^2\).

Step 3: Add the \(c\) terms

Add the \(c\) terms: \(-3c\) (since \(14c^2\) was written as \(14c^2 + 0c\)) does not add to any other \(c\) term, so it remains as \(-3c\).

Step 4: Add the constant terms

Add the constant terms: \(-5 - 1 = -6\).

Step 5: Write the Final Result

Combine all the sums from the previous steps to write the polynomial result:\(12c^2 - 3c - 6\).

Key Concepts

Vertical Form

When dealing with polynomial addition, using the vertical form can simplify the process greatly. This method is similar to how we add numbers in arithmetic, vertically aligning elements based on their place value. For polynomials, the place value analogy translates into terms. Specifically, when we talk about aligning terms, we mean aligning them by their degree or the power of the variable.
For example, a polynomial like

• \(-2c^2 - 3c - 5\)

and another like

• \(14c^2 - 1\)

can be structured such that terms of the same degree are placed in corresponding columns. This vertical stacking makes it easy to add together the coefficients of like terms. It's a clear and organized approach that reduces the chance of making errors during calculation.

Like Terms

Understanding the concept of "like terms" is fundamental in polynomial operations. Like terms are terms that contain the same variables raised to the same power. Consider the polynomial terms in our example.

• \(-2c^2\) and \(14c^2\) are like terms because they both contain \(c^2\).
• \(-3c\) stands alone in the given polynomials as another kind of term based on its variable \(c\).

Constant terms like \(-5\) and \(-1\) are also considered like terms because they lack variables entirely. Identifying like terms allows you to combine them effectively, simplifying expressions in the polynomial.

Column Alignment

Column alignment is crucial when adding polynomials vertically. Each column represents a set of like terms, whether they're constants, terms with one variable, or polynomials with higher degrees of that variable.
For instance, in our problem:

• The \(c^2\) terms are aligned in one column.
• The \(c\) term from the first polynomial aligns with a zero placeholder for clarity since there is no \(c\) term in the second polynomial.
• Constants form the final column.

By aligning the polynomials this way, you'll ensure that each set of like terms is combined correctly. Using a zero, like in \(0c\), as a placeholder helps maintain accurate alignment even when a polynomial lacks a particular term. This prevents mishaps where terms might be misaligned or forgotten.

Arithmetic Operations with Polynomials

Performing arithmetic operations with polynomials, like addition in our example, follows straightforward arithmetic rules once terms are aligned correctly. Standard arithmetic operations apply:

• Add the coefficients of like terms.
• Combine constants as you normally would with basic numbers.

For example, in our exercise, you add:

• \(-2c^2\) and \(14c^2\) to get \(12c^2\).
• \(-3c\) remains as it is since there is no like term in the second polynomial.
• Add the constants \(-5\) and \(-1\) to obtain \(-6\).

Once each set of like terms is added, you combine the results to form the final simplified polynomial. This process illustrates how arithmetic operations are applied to polynomials in a structured manner when using vertical form and proper alignment techniques.