Problem 9 - Algebra and Trigonometry

Question

In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. $$ \left(-6 x^{3}+5 x^{2}-8 x+9\right)+\left(17 x^{3}+2 x^{2}-4 x-13\right) $$

Step-by-Step Solution

Answer

The polynomial in standard form is \( 11x^3 + 7x^2 - 12x - 4 \), and its degree is 3.

Step 1: Identify Like Terms

We can start off by highlighting like terms in the two polynomials. Like terms are the ones that have the same variable raised to the same power in this case. Those terms are: \( -6x^3 \) and \( 17x^3 \), \( 5x^2 \) and \( 2x^2 \), \( -8x \) and \( -4x \), and finally the constants \( 9 \) and \( -13 \).

Step 2: Add Like Terms

Add the coefficients of the like terms from both polynomials to get new coefficients. These operations are: \( -6x^3 + 17x^3 \), \( 5x^2 + 2x^2 \), \( -8x - 4x \) and \( 9 - 13 \). This results in a new polynomial which is: \( 11x^3 + 7x^2 - 12x - 4 \).

Step 3: Write in Standard Form and Indicate Degree

The resulting polynomial is already in standard form because the terms are arranged in descending order of their degrees. The polynomial is: \( 11x^3 + 7x^2 - 12x - 4 \). The degree of this polynomial would be the exponent of the term with the highest exponent, which is 3 in this case. Thus, the degree of the polynomial is 3.

Key Concepts

Polynomial Operations

Understanding polynomial operations is essential for working with algebraic expressions. When adding polynomials, like the problem at hand \( (-6 x^{3}+5 x^{2}-8 x+9) + (17 x^{3}+2 x^{2}-4 x-13) \), the process involves combining like terms, which are terms that have the same variable to the same power.

To perform the addition, simply sum up the coefficients (numbers in front of the variables) of the like terms. For instance, we combine \( -6x^3 \) with \( 17x^3 \) to get \( 11x^3 \). This is done for each set of like terms across the entire polynomial expression.

Adding Step by Step

• First, identify all the like terms.
• Then, add the coefficients of these like terms.
• Lastly, rewrite the expression with the updated coefficients.

Through this simple operation, you can easily add any two polynomials to arrive at a new, combined polynomial.

Standard Form of a Polynomial

A polynomial in standard form is written with the terms arranged in descending order by degree, which is the exponent of the variable.

For example, the polynomial \( 11x^3 + 7x^2 - 12x - 4 \) is in standard form because the term with the highest degree comes first \(x^3\), followed by the term with the next lower degree \(x^2\), and so on down to the constant term, which has a degree of 0.

Writing polynomials in standard form is crucial because it makes it easier to read and understand the most significant part of the polynomial, and is also essential for performing polynomial operations such as addition, subtraction, multiplication, and division.

Degree of a Polynomial

The degree of a polynomial is the highest power of the variable in the polynomial when it is written in standard form. Knowing the degree of a polynomial helps us understand the behavior of the graph of the polynomial function, particularly the number of possible zero values or roots.

In the example \( 11x^3 + 7x^2 - 12x - 4 \), the highest power is 3, coming from the term \(11x^3\). Therefore, the degree of this polynomial is 3. The degree can give us a quick idea about the number of roots, possible extremas (maximums or minimums), and the end behavior of the polynomial graph.

Like Terms in Polynomials

In polynomials, like terms are terms that have exactly the same variables raised to the same powers. These are the only types of terms that can be combined through addition or subtraction.

In the given exercise, the like terms identified are terms that have the variable \(x\) raised to the same powers. For instance, \( -6x^3 \) and \( 17x^3 \) are like terms because both have the variable \(x\) raised to the third power. It is important to remember that constants are also considered to be like terms because they can be viewed as the variable raised to the power of 0. This property of like terms is pivotal when simplifying and combining polynomial expressions.