Determine if a Polynomial -10x^6-2x^5+4x^4+7x^3-9x^2+5.5x-1
The problem is asking for an analysis of the given polynomial, which is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, the polynomial is in one variable, x, and consists of several terms each with different powers of x. The terms have both positive and negative coefficients, and the powers range from 6 down to 0 (the constant term). The problem may involve checking certain properties of the polynomial, such as its degree (which is the highest power of the variable x that appears with a non-zero coefficient), its roots (values of x that make the polynomial equal to zero), or other features specific to polynomials. The polynomial in question is:
-10x^6 - 2x^5 + 4x^4 + 7x^3 - 9x^2 + 5.5x - 1.
$- 10 x^{6} - 2 x^{5} + 4 x^{4} + 7 x^{3} - 9 x^{2} + 5.5 x - 1$
Answer
Solution:
Step:1
A polynomial is an algebraic expression that consists of a sum of several terms, each term including a variable raised to a non-negative integer exponent. Polynomials do not include:
Negative or fractional powers of variables (e.g., $2x^{-2}$, $x^{1/2}$).
Variables in the denominator (e.g., $\frac{1}{x}$, $\frac{1}{x^2}$).
Radicals involving variables (e.g., $\sqrt{x}$, $\sqrt[3]{x}$).
Non-algebraic functions such as trigonometric functions, absolute values, or logarithms.
Step:2
Examine the given expression to see if it violates any of the polynomial rules.
The expression adheres to all the rules.
Step:3
Conclude whether the given expression qualifies as a polynomial.
The expression is indeed a polynomial.
Knowledge Notes:
Polynomials are expressions in mathematics that consist of variables (also known as indeterminates) and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Here are some key points about polynomials:
Terms and Degree: A polynomial is made up of one or more terms. Each term is a product of a constant coefficient and a variable raised to a power. The degree of a polynomial is the highest power of the variable in the polynomial.
Standard Form: A polynomial is often written in standard form when its terms are arranged in descending order of their degrees.
Types of Polynomials: Based on the number of terms, polynomials can be classified as monomials (one term), binomials (two terms), trinomials (three terms), or more generally as polynomials with more terms.
Operations on Polynomials: Polynomials can be added, subtracted, multiplied, and divided (except by zero). Division of polynomials may not always result in a polynomial.
Special Cases: Polynomials must not include variables with negative or fractional exponents, variables in the denominator, or variables under a radical. They also must not include functions such as sine, cosine, absolute value, or logarithms as part of the terms.
Roots/Zeros of Polynomials: The values of the variable that make the polynomial equal to zero are called roots or zeros. The Fundamental Theorem of Algebra states that a polynomial of degree $n$ has exactly $n$ roots, counting multiplicity.
Graphs of Polynomials: The graph of a polynomial function is a smooth, continuous curve. The degree of the polynomial can give us information about the number of turning points the graph can have.
In the context of the given problem, the expression $-10x^6 - 2x^5 + 4x^4 + 7x^3 - 9x^2 + 5.5x - 1$ meets all the criteria of a polynomial: it has variables with non-negative integer exponents, no variables in the denominator, no radicals, and no non-algebraic functions. Thus, it is confirmed to be a polynomial.
