Problem 25 - Algebra Form and Function

Question

List the nonzero coefficients of the polynomials. $$ 3 u^{4}+6 u^{3}-3 u^{2}+8 u+1 $$

Step-by-Step Solution

Answer

Answer: The non-zero coefficients of the given polynomial are 3, 6, -3, 8, and 1.

Step 1: Identify the terms in the polynomial

In the given polynomial, there are 5 terms: $$ 3u^4, \quad 6u^3, \quad -3u^2, \quad 8u, \quad 1 $$

Step 2: List the non-zero coefficients

Now, we just need to list the coefficients of the terms we identified in Step 1: $$ 3, \quad 6, \quad -3, \quad 8, \quad 1 $$ So, the non-zero coefficients are 3, 6, -3, 8, and 1.

Key Concepts

Nonzero Coefficients

Understanding nonzero coefficients is crucial in working with polynomials. Coefficients are the numerical part of terms in an expression. When we refer to nonzero coefficients, we mean those coefficients that are not equal to zero. They play a significant role as they affect the value of the polynomial. Without them, the usefulness of our polynomial terms would be limited.
For example, in the polynomial expression \(3u^4 + 6u^3 - 3u^2 + 8u + 1\), each term has a nonzero coefficient:

• Coefficient of \(u^4\) is 3
• Coefficient of \(u^3\) is 6
• Coefficient of \(u^2\) is -3
• Coefficient of \(u\) is 8
• Constant term is 1

In simpler terms, the expression cannot be simplified further by removing these coefficients, as each one contributes to the overall equation.

Polynomial Terms

Polynomials are made up of multiple terms. Each term in a polynomial is a product of a coefficient and a variable raised to a power. A term's degree is determined by its exponent, which tells us the highest power of the variable in that term.
Consider the polynomial given: \(3u^4 + 6u^3 - 3u^2 + 8u + 1\). This expression has five distinct polynomial terms:

• \(3u^4\) - The first term, with a degree of 4
• \(6u^3\) - The second term, with a degree of 3
• \(-3u^2\) - The third term, with a degree of 2
• \(8u\) - The fourth term, with a degree of 1
• \(1\) - The constant term, with a degree of 0

Understanding each term's degree and how terms combine is important for simplifying and evaluating polynomials. It helps us order them either in standard form or combine similar terms.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that together create meaningful mathematical statements. They form the basis for algebra and calculus. A polynomial is one type of algebraic expression where variables are raised to whole number exponents and combined by addition or subtraction.
The expression \(3u^4 + 6u^3 - 3u^2 + 8u + 1\) is an excellent example of a polynomial. It contains:

• Variables - Represented by 'u'
• Coefficients - The numbers next to variables
• Operators - Plus and minus signs which separate terms

Understanding algebraic expressions requires recognizing these components and knowing how they function together. This allows us to manipulate and solve algebraic problems efficiently.