Problem 45 - Elementary and Intermediate Algebra

Question

Use vertical form to add the polynomials. $$ \begin{array}{r} {z^{3}+6 z^{2}-7 z+16} \\ {9 z^{3}-6 z^{2}+8 z-18} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is \(10z^3 + z - 2\).

Step 1: Align the Terms

First, write the given polynomials in a vertical format, ensuring that you align like terms (terms with the same power of \(z\)) in columns. \[\begin{array}{r} {z^{3}+6 z^{2}-7 z+16} \ {+9 z^{3}-6 z^{2}+8 z-18} \ \hline\end{array}\]

Step 2: Add the Coefficients of \(z^3\) Terms

Add the coefficients of the \(z^3\) terms from both polynomials. Here, the coefficients are 1 (from \(z^3\)) and 9 (from \(9z^3\)).Sum: \[1 + 9 = 10\]

Step 3: Add the Coefficients of \(z^2\) Terms

Next, add the coefficients of the \(z^2\) terms. The coefficients are 6 (from \(6z^2\)) and -6 (from \(-6z^2\)).Sum: \[6 + (-6) = 0\]

Step 4: Add the Coefficients of \(z\) Terms

Now, add the coefficients of the \(z\) terms, which are -7 (from \(-7z\)) and 8 (from \(8z\)).Sum: \[-7 + 8 = 1\]

Step 5: Add the Constant Terms

Lastly, add the constant terms, 16 and -18.Sum: \[16 + (-18) = -2\]

Step 6: Write the Resultant Polynomial

Combine the results from each addition step to form the final polynomial: \[10z^3 + 0z^2 + 1z - 2\]Since the \(z^2\) term has a coefficient of 0, it can be omitted from the expression.Final result:\[10z^3 + z - 2\]

Key Concepts

Vertical Form

When you add polynomials, using a vertical form can make the process much easier. Think of adding polynomials in the same way as adding large numbers vertically. Each kind of term, like constant terms, first-degree terms, or squared terms, has to be lined up with similar terms. This ensures that you only add like terms together.

Arrange the terms of each polynomial so that similar terms are in the same vertical line. For example, if you have two polynomials including terms like \(z^3\), \(z^2\), \(z\), and constants, each type should be stacked above or below their corresponding pair. This layout facilitates straightforward addition, just like how you'd sum up digits in addition problems.

Overall, using vertical form allows for a clear, organized overview of your work and helps prevent errors by ensuring proper term alignment.

Aligning Polynomials

Aligning polynomials is a crucial step in polynomial addition. It involves placing each polynomial's terms so that terms with like powers appear directly above or below each other.

Here's how you can align polynomials effectively:

• Start by writing each polynomial on a separate line, one beneath the other.
• Ensure that each polynomial's terms are ordered by descending powers of the variable (in this case, \(z\)).
• Leave space for any missing terms so that empty spots can be accounted for in the alignment.

This systematic arranging makes it easier to add the coefficients of like terms without confusion. Think of it as aligning digits in basic arithmetic to ensure you correctly calculate the result.

Like Terms

Like terms in a polynomial are terms that share the exact same variable raised to the same power. In other words, the structure is identical except for the coefficient.

When adding polynomials, it is crucial to identify and add only like terms:

• Terms like \(z^3\) and \(z^3\) are like terms because they both involve \(z\) raised to the same power.
• However, terms like \(z^2\) and \(z\) are not like terms because their powers are different (2 compared to 1).

Adding different terms by their highest or lowest power would be incorrect, resulting in an inaccurate polynomial. Hence, ensure that each term is summed with its corresponding like term.

Polynomial Coefficients

Polynomial coefficients are the numbers in front of the variable terms. These are the values that will vary when manipulating polynomials, such as when adding terms.

The role of coefficients is paramount when adding polynomials:

• For each power of the variable, add only the coefficients. For example, when you add \(z^3\) terms, you're really adding the numbers in front of each \(z^3\), like \(1\) and \(9\) to make \(10z^3\).
• It's important to maintain the same order of terms across polynomials so that addition is straightforward, leading to an accurate final polynomial.

Remember, in cases where a certain term is missing from a polynomial (like the \(z^2\) term disappearing when coefficients sum to zero), you simply do not write that term in the resultant polynomial because its coefficient is zero.