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Chapter 0: Problem 97

What is a polynomial in \(x ?\)

Short Answer

Expert verified
A Polynomial in 'x' is an expression which consists of terms with variables and coefficients combined using only addition, subtraction, and multiplication, and not division by a variable. The exponents of 'x' in the terms are positive integers, or zero.

Step by step solution

01

Understanding a Polynomial

A Polynomial is an expression consisting of variables, coefficients, and exponents, that are combined using addition, subtraction, and multiplication, but no division by a variable. A polynomial in 'x' is specifically an expression in which 'x' is the variable.
02

Constructing a Polynomial

A polynomial in 'x' can have any number of terms, and its power is determined by the highest exponent of 'x'. For example, \(4x^2 - 3x + 2\) and \(5x^7 - x^3 + 2x - 4\) are polynomials in 'x', where the first has degree 2 and the second has degree 7. The numerical coefficients can be any real numbers.
03

Non-Polynomial Expressions

Not all expressions with 'x' are polynomials. Expressions that include division by a variable, negative exponents, fractional exponents, or root symbols are not considered polynomials. For example, \(1/x\), \(x^{-2}\), and \(\sqrt{x}\) are not polynomials in 'x'.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$\frac{2 x-1}{x-7}+\frac{3 x-1}{x-7}-\frac{5 x-2}{x-7}=0$$

Write a quadratic equation in general form whose solution set is \(\\{-3,5\\}\).

Rationalize the numerator. $$\frac{\sqrt{x}+\sqrt{y}}{x^{2}-y^{2}}$$

This will help you prepare for the material covered in the next section. If 6 is substituted for \(x\) in the equation $$2(x-3)-17=13-3(x+2)$$ is the resulting statement true or false?

Exercises \(142-144\) will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$ 2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right) $$
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