Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ \left(x^{3}+3 x^{2}+2\right)+\left(x^{2}-4 x+4\right) $$

Short answer

\[ x^{3} + 4x^{2} - 4x + 6 \]

Step-by-step solution

- Write down the given polynomials

List the two polynomials that need to be added: \[ \text{First Polynomial: } x^{3} + 3x^{2} + 2 \ \text{Second Polynomial: } x^{2} - 4x + 4 \]

- Combine like terms

Add the like terms from both polynomials. This involves combining terms with the same degree: \[ \begin{aligned} &\text{Combine } x^{3}: \ &\text{Combine } x^{2}: \ &\text{Combine } x: \ &\text{Combine the constant terms:} \ & x^{3} \ + (3x^{2} + x^{2}) \ = 4x^{2} \ - 4x \ + (2 + 4) = 6 \ \text{So, the result is: } x^{3} + 4x^{2} - 4x + 6 \]

- Write the polynomial in standard form

Arrange the polynomial in descending order of the exponents:\[ x^{3} + 4x^{2} - 4x + 6 \ \text{This is the polynomial in standard form} \]

Key concepts

Adding Polynomials

Adding polynomials is much like adding regular numbers, but you need to pay close attention to the variables and their exponents. To add polynomials, you simply combine the coefficients (the numbers in front of the variables) of like terms (terms that contain the same variable raised to the same power). For example, when adding the two polynomials:

• First Polynomial: \(x^{3} + 3x^{2} + 2\)
• Second Polynomial: \(x^{2} - 4x + 4\)

You look at each term separately. Add the coefficients of terms with corresponding variables. For example, there is only one \(x^{3}\) term, so it remains \(x^{3}\). For the \(x^{2}\) terms, you add the coefficients: \(3x^{2} + x^{2} = 4x^{2}\). Continue this process for all terms to combine them appropriately.

Combining Like Terms

Combining like terms is an essential step when adding polynomials. Like terms are terms that have the same variable and the same exponent. For instance, in the given exercise, you combine terms as follows:

• Combine the \(x^{3}\) terms: Only \(x^{3}\) from the first polynomial is present.
• Combine the \(x^{2}\) terms: \(3x^{2} + x^{2} = 4x^{2}\) by adding the coefficients (3 + 1). This gives the total of \(4x^{2}\).
• Combine the \(x\) terms: Only the \(-4x\) term is present from the second polynomial.
• Combine the constant terms: \(2 + 4 = 6\).

This process simplifies the expression into another polynomial, making it easier to handle.

Standard Form of a Polynomial

After combining like terms, writing the polynomial in standard form is the next step. Standard form means arranging the terms from highest to lowest degree. For the polynomial from the exercise: \(x^{3} + 4x^{2} - 4x + 6\), the terms are already ordered this way. Here is a quick refresher:

• The degree of a term is the exponent of the variable. For example, \(x^{3}\) has a degree of 3.
• In our polynomial, the terms are \(x^{3} + 4x^{2} - 4x + 6\).
• We order it as: \(x^{3}\) (highest degree), \(4x^{2}\), \(-4x\), and \(6\) (constant term).

By following these steps, you ensure your polynomial is properly simplified and presented in a clear, organized manner, making further operations or evaluations easier.