Problem 118 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether the expression is a polynomial. If it is, write the polynomial in standard form. $$3 x-10 x^{2}+1$$

Step-by-Step Solution

Answer

Yes, the given expression is a polynomial and it's standard form is \(-10 x^{2} + 3x + 1\).

Step 1: Identify the Polynomial

A Polynomial is an expression that can be written in the form \(a_n x^n + a_{n-1} x^{n-1}+ ... + a_2 x^2 + a_1 x^1 + a_0\), where each \(a\) is a real number and \(n\) should be a nonnegative integer. Here in the given expression \(3 x - 10 x^{2} + 1\), each term is a product of a constant and a variable raised to a non-negative integer. Which means the given expression follows the definition of a polynomial.

Step 2: Write the Polynomial in Standard Form

To write a polynomial in standard form, we arrange the terms so that the power of \(x\) decreases from left to right. Therefore, we rewrite \(3 x - 10 x^{2} + 1\) as \(-10 x^{2} + 3x + 1\) which is the standard form of the given polynomial.

Key Concepts

Polynomials in Mathematics

Polynomials play a pivotal role in mathematics, forming the foundation for more complex topics in algebra and calculus. A polynomial is a mathematical expression that comprises variables (also known as indeterminates), coefficients, and non-negative integer exponents of those variables. The simplest polynomials consist of a single term, like 3x, and are known as monomials. When two or more terms are combined, we get binomials, such as x^2 + 2x, or trinomials like x^2 + 2x + 1. More generally, any expression with a sum of multiple terms is known as a polynomial.

The greatest exponent of a variable in a polynomial dictates the degree of the polynomial. For instance, a quadratic polynomial such as ax^2 + bx + c has a degree of two, while a cubic polynomial like ax^3 + bx^2 + cx + d has a degree of three. Understanding the structure of polynomials is essential as they can model various real-life occurrences, from calculating areas and volumes to describing the trajectory of objects.

Writing Polynomials

When writing polynomials, it's not just about laying down numbers and variables haphazardly. There is a standard form to follow that helps keep expressions organized and facilitates further mathematical operations. The standard form of a polynomial requires that the terms be arranged in descending order of their degrees, which means the term with the highest exponent should be placed first, followed by the term with the next lower exponent, and so on, until the constant term is at the end.

For example, rewriting the expression 3x - 10x^2 + 1 in standard form leads us to -10x^2 + 3x + 1. Notice how the highest power, that is the squared term, leads the expression, with exponents then decreasing from there. Adhering to this form not only makes it easier to read and compare polynomials but also sets a consistent framework for addition, subtraction, and multiplication of polynomial expressions.

Polynomial Expression

A polynomial expression is the actual string of symbols that represents the polynomial itself. It's composed of terms, where each term is a product of a real number coefficient (like 3 or -10 in the expression 3x - 10x^2 + 1) and powers of variables (like x^2 or x). The beauty of a polynomial expression lies in its versatility; it can simplify complex scenarios into manageable equations that can be solved using algebraic rules.

One critical characteristic of polynomial expressions is that the powers of the variable must be whole numbers—that is, no negative exponents, fractional exponents, or variables in the denominator. For example, x^-1 or 1/x are not considered part of a polynomial expression. Hence, identifying correctly formed polynomial expressions is crucial in solving and manipulating these mathematical tools effectively.