Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation(s) and write the resulting polynomial in standard form.\(z^{2}\left(2 z^{2}+3 z+1\right)\)

Step-by-Step Solution

Answer

The resulting polynomial in standard form is \(2z^{4} + 3z^{3} + z^{2}\)

Step 1: Multiply every term in the Polynomial

The first step is to multiply every term in the polynomial \(2 z^{2}+3 z+1\) by the term \(z^{2}\). This gives: \(z^{2} * 2z^{2} = 2z^{4}\), \(z^{2} * 3z = 3z^{3}\), \(z^{2} * 1 = z^{2}\). So the resulting polynomial is \(2z^{4} + 3z^{3} + z^{2}\)

Step 2: Write the Polynomial in Standard Form

Now, write the resulting polynomial in standard form which means to write the polynomial with the terms ordered by the degree from largest to smallest. Since the resulting polynomial \(2z^{4} + 3z^{3} + z^{2}\) is already in standard form, no change is required in this step.

Key Concepts

Polynomial Multiplication

Polynomial multiplication involves multiplying two or more polynomials together by distributing each term in one polynomial to every term in the other polynomial. In our exercise, we had to multiply the polynomial \( z^2 \) by \( 2z^2 + 3z + 1 \). This is done by:

• Multiplying \( z^2 \) by each term in \( 2z^2 + 3z + 1 \).
• First, multiply \( z^2 \) and \( 2z^2 \) to get \( 2z^4 \).
• Then, multiply \( z^2 \) and \( 3z \) to obtain \( 3z^3 \).
• Lastly, multiply \( z^2 \) and \( 1 \) to get \( z^2 \).

As a result of these operations, you combine all the terms to get the polynomial \( 2z^4 + 3z^3 + z^2 \). Remember to perform distribution carefully to avoid errors in your calculations.

Standard Form of a Polynomial

The standard form of a polynomial means placing the polynomial terms in descending order of their degrees. Once you complete the polynomial multiplication, ensure that each term is ordered from highest to lowest degree.

After performing the multiplication, the polynomial \( 2z^4 + 3z^3 + z^2 \) is obtained. It's already in standard form:

• The term \( 2z^4 \) is highest in degree and comes first.
• Next is \( 3z^3 \), a degree lower, and follows \( 2z^4 \).
• The term \( z^2 \) is the last as it has the lowest degree of the terms present.

By following this order, anyone reading or using the polynomial can easily discern its terms based on their degree.

Degree of a Polynomial

The degree of a polynomial is determined by the highest power of the variable present in the polynomial. In polynomial operations, identifying the degree helps in ordering terms and understanding the behavior of the polynomial.

For the polynomial \( 2z^4 + 3z^3 + z^2 \), the degree is 4. This is because the term with the highest exponent is \( 2z^4 \), where the exponent (4) indicates the degree.

• The degree tells us the maximum number of roots the polynomial can have.
• It helps calculate the end behavior when looking at graphs of the polynomial function.
• In expressions and equations, it indicates the level of demand in computation complexity.

Understanding the degree is essential as it forms the basis for nearly all polynomial analyses and operations.