Problem 42 - Elementary and Intermediate Algebra

Question

Use vertical form to add the polynomials. $$ \begin{array}{l} {6 x^{3}-4 x^{2}+7} \\ {7 x^{3}+9 x^{2}+12} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum of the polynomials is \(13x^3 + 5x^2 + 19\).

Step 1: Arrange Polynomials Vertically

Write the polynomials in a vertical form, aligning the like terms in columns. This means putting the terms with the same degree in the same column:\( \begin{array}{rl} & 6x^3 - 4x^2 + 0x + 7 \+ & 7x^3 + 9x^2 + 0x + 12 \\hline\end{array} \)

Step 2: Add the Cubic Terms

Add the cubic terms from both polynomials. The cubic terms are in the first column:\(6x^3 + 7x^3 = 13x^3\)

Step 3: Add the Quadratic Terms

Next, add the quadratic terms, which are in the second column:\(-4x^2 + 9x^2 = 5x^2\)

Step 4: Add the Linear Terms

In this problem, there are no linear terms in either polynomial, but it's good practice to recognize their alignment as zeros:\(0x + 0x = 0x\)

Step 5: Add the Constant Terms

Finally, add the constant terms from both polynomials, found in the last column:\(7 + 12 = 19\)

Step 6: Combine All Results

Combine all the results from the previous steps to write the final sum of the polynomials:The sum of the polynomials is:\(13x^3 + 5x^2 + 0x + 19\) which simplifies to \(13x^3 + 5x^2 + 19\).

Key Concepts

Understanding the Vertical Method

The vertical method is a straightforward and structured approach to adding polynomials. It involves writing each polynomial term-by-term in a column format, aligning terms with the same degree. This structure enables you to clearly see and focus on adding like terms, much like adding numbers with similar place values in traditional arithmetic. When using the vertical method, ensure that:

• The terms are written in descending order of their degrees, from highest to lowest.
• Each type of term - cubic, quadratic, linear, and constant - is stacked in its corresponding column.

This visualization makes it easy to track which terms are added together, reducing error and ensuring clarity in operations.

Identifying and Combining Like Terms

In polynomial addition, like terms are those that have the same variable raised to the same power. For example, consider the terms \(6x^3\) and \(7x^3\); both are cubic terms, and therefore, they can be added together. Identifying and combining like terms accurately is crucial:

• Only terms with the exact same variables and exponents can be combined. This ensures that the integrity of the polynomial is maintained.
• Think of it like combining apples with apples, and not with oranges. It ensures coherent results without altering the final polynomial expression.

The process simplifies the polynomial and creates a smooth path to finding the answer without mistakes.

Adding Cubic Terms

Cubic terms are polynomial terms where the variable is raised to the third power, like \(6x^3\) and \(7x^3\). When adding cubic terms, simply add their coefficients—the numbers in front of the variables. In our exercise:

• The coefficients 6 and 7 are added together.
• This results in a new coefficient of 13, producing the term \(13x^3\).

Cubic terms significantly influence the shape of a polynomial graph as they form the highest degree's terms. Ensuring these are added correctly is essential for the accuracy of polynomial sums.

Adding Quadratic Terms

Quadratic terms refer to those in which the variable is squared, such as \(-4x^2\) and \(9x^2\). Similar to adding cubic terms, the coefficients of the quadratic terms are added together:

• The quadratic term coefficients are -4 and 9.
• Adding the coefficients results in \(5x^2\).

Quadratic terms define the parabolic sections in polynomial graphs and checking their coefficients carefully can ensure precise solutions. This step is vital for capturing the middle portion of any polynomial expression accurately.