Problem 10 - Algebra and Trigonometry

Question

In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$ \left(-7 x^{3}+6 x^{2}-11 x+13\right)+\left(19 x^{3}-11 x^{2}+7 x-17\right) $$

Step-by-Step Solution

Answer

The resultant polynomial in standard form is \(12x^3 - 5x^2 - 4x - 4\) and its degree is 3.

Step 1: Add Corresponding Terms

The corresponding terms from the two polynomials are added to produce the resultant polynomial: \[ (-7x^3 + 19x^3) + (6x^2 - 11x^2) + (-11x + 7x) + (13 - 17) \]

Step 2: Simplify

After adding each pair of corresponding terms from the two polynomials, the simplification becomes: \[ 12x^3 - 5x^2 - 4x - 4 \]

Step 3: Identify the Degree

The degree of the polynomial is the highest power of the variable in the polynomial. Here, the highest power of the variable x is 3. Thus, the degree of the polynomial is 3.

Key Concepts

Adding Polynomials

When it comes to adding polynomials, the process is quite straightforward. Polynomials are algebraic expressions that consist of terms, which are the variables raised to a power and multiplied by a coefficient (for example, in the term \(7x^3\), 7 is the coefficient and \(x^3\) is the variable raised to the third power).

Here's how to add polynomials effectively:

• First, identify like terms. These are terms that have the exact same variable raised to the same power.
• Then, simply add the coefficients of these like terms together.
• Leave the exponents unchanged because mathematical rules entail that we don't alter the exponents when adding or subtracting terms.

The exercise given above demonstrates this process with two polynomials. By following the steps of adding the corresponding terms, we achieve a new polynomial that accurately represents the sum of the original polynomials.

Standard Form of a Polynomial

A polynomial should ideally be written in standard form, which is important for clarity and often required for textbook exercises. Writing a polynomial in standard form includes two key steps:

• First, arrange the terms so that the powers of the variable are descending, starting with the highest power first. This means if your polynomial has terms with the variables \(x^3\), \(x^2\), \(x\), and a constant, they should be arranged in that order.
• Second, ensure that the coefficients are positioned right in front of the respective variables. For instance, \(2x^2\) should be written as such and not \((x^2)2\).

Your resulting polynomial in standard form will allow anyone reading it to easily understand the hierarchy of terms based on their degree, thus simplifying further operations or evaluations of the polynomial.

Degree of a Polynomial

The degree of a polynomial is a fundamental concept that determines the highest power of the variable in the polynomial. It's quite literally a straightforward counting task -- you just look for the term with the largest exponent. The degree gives insight into the behavior of the polynomial, especially concerning its graph and the nature of its roots.

For instance, in the sum of the polynomials presented in our original exercise, we identify the highest power of \(x\) which is 3 (coming from the term \(12x^3\)). Therefore, we say the polynomial has a degree of 3. Knowing the degree of a polynomial helps us predict the number of roots and the maximum number of turns in its graph, which can be essential for graphing and understanding the polynomial's behavior.