What is the Degree One Polynomial: Debunking the Myths

In this article, we will delve into the polynomial expression 3a4bx^3-a-4x^25a2b-4x5 and explore the concept of a degree one polynomial. We will also address common misconceptions and provide a step-by-step explanation to arrive at the correct solution.

Misconceptions and Clarifications

There seems to be a prevalent belief that not all polynomials meeting the criteria of a degree one polynomial are considered legitimate. However, in this case, the problem explicitly states that 3a4bx^3-a-4x^25a2b-4x5 should be a degree one polynomial. Let's analyze the expression and resolve the doubts surrounding this claim.

Second Degree Polynomial Terms: The terms 3a4bx^3 and -4x^25a2b-4x5 indicate that there are second degree terms in the expression. For the polynomial to be a degree one polynomial, these second degree terms must vanish. This implies that the coefficient of x^3 (3a4b) must be zero, and similarly, the coefficient of x^2 (a-4) must also be zero.

Resolving Coefficients: Let's begin by setting the coefficient of x^3 to zero:

3a4b 0

From this equation, we can deduce that 3a -4b. Next, let's set the coefficient of x^2 to zero:

a - 4 0

Solving this equation, we find:

a 4

Substituting a 4 into the equation 3a -4b, we get:

3(4) -4b

12 -4b

b -3

Identifying the First Degree Term

With the values a 4 and b -3, we can rewrite the original polynomial expression. The constant term (the eighth term in the expression) remains as is, and the other terms that are not of degree one can be set to zero. Thus, the first degree polynomial is:

5x^4 2x-3 -4x5

Simplifying this, we find the coefficient of the linear term:

5(4) 2(-3) -4 20 - 6 - 4 10

So, the final degree one polynomial is:

1x 10

Conclusion: Addressing Common Misunderstandings

Some may argue that this polynomial is not a first degree polynomial because of the presence of higher degree terms. However, the problem definition states that this expression is a degree one polynomial. Thus, the terms of degree higher than one must be zero, as per the rules of polynomial degrees.

Additionally, it is essential to understand that the degree of a polynomial refers to the highest degree of its constituent terms. Hence, a polynomial is considered a degree one polynomial as long as the coefficient of the highest degree term (in this case, x to the first power) is non-zero, while all higher degree terms are zero.

In conclusion, the degree one polynomial in this context is 1x 10. This highlights the importance of adhering to the given problem statement and correctly interpreting the behavior of polynomial terms.