Problem 47 - Elementary and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The sum of the polynomials is \(-x^3y^2 + 4x^2y + 5x + 6\).

Step 1: Align the Polynomials

Write the polynomials in a way that aligns the like terms in a vertical arrangement. Make sure that the terms with the same degree are directly one above the other. If a polynomial does not have a certain degree, leave that column blank.\[\begin{array}{r}-3x^3y^2 + 4x^2y - 4x + 9 \2x^3y^2 + 0x^2y + 9x - 3\end{array}\]

Step 2: Add Like Terms Starting from Highest Degree

Add the coefficients of the like terms, starting from the highest degree term to the constant term. Perform addition column by column.1. \(-3x^3y^2 + 2x^3y^2 = -1x^3y^2\)2. \(4x^2y + 0x^2y = 4x^2y\)3. \(-4x + 9x = 5x\)4. \(9 - 3 = 6\)

Step 3: Write the Resultant Polynomial

Combine all the results from the addition of like terms to get the final polynomial. Ensure you write the polynomial in descending order of degree.\[-x^3y^2 + 4x^2y + 5x + 6\]

Key Concepts

Vertical Form

Vertical form is a common technique for adding polynomials, similar to the method used for adding numbers. In this approach, each polynomial is written out from top to bottom, with terms aligned in columns according to their degrees. This visual setup helps ensure each term is easily identifiable and prevents errors during calculation.

• Write the polynomials one below the other, ensuring the terms of similar degree are lined up vertically.
• Fill in any missing degrees with zero terms to maintain alignment.

This method simplifies the addition process by visually grouping like terms for direct comparison and calculation. Think of it as stacking the polynomials for clarity.

Like Terms

Like terms are terms in a polynomial that have the same variables raised to the same powers. Identifying like terms is crucial for correctly adding polynomials.When combining terms:

• Ensure each group of like terms is aligned vertically to make addition straightforward.
• Only the coefficients of like terms are added or subtracted, not the variables.

For example, in the polynomial expression \(-3x^3y^2 + 2x^3y^2\), both terms are like terms because they contain the same variables, \(x^3y^2\). The operation then becomes solely about the coefficients, leading to \(-1x^3y^2\). Recognizing like terms is a key skill in simplifying polynomials.

Coefficients

Coefficients in polynomials are the numerical part of terms. They play a crucial role in operations like addition and subtraction.Here's how you manage coefficients in polynomial addition:

• Focus only on the coefficients when you add or subtract like terms.
• Combine them by simple arithmetic addition or subtraction.

For example, in adding \(-4x + 9x\), only consider the coefficients \(-4\) and \(9\). The terms combined result in \(5x\). Coefficients tell us how many groups of the variable terms we have, dictating the magnitude of each term in the polynomial.

Degree of Polynomial

The degree of a polynomial refers to the highest power of the variable(s) present in the polynomial. The degree indicates both the complexity and the behavior of the polynomial function.In context:

• Identify the degree by looking for the term with the highest sum of exponents.
• Degrees influence how you align terms in vertical form; terms are arranged from highest to lowest degree.

In our example, the polynomial \(-3x^3y^2\) comes with a degree of 5, from adding the exponents of \(x\) and \(y\). Writing polynomials in order of descending degree ensures difficulty in terms is managed, allowing for easier arithmetic handling.