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Chapter 0: Problem 97

What is a polynomial in \(x ?\)

Short Answer

Expert verified
A polynomial in \(x\) is a mathematical expression that can be presented in the form \(p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\), where \(n\) is a non-negative integer, and \(a_0, a_1, ..., a_n\) are the coefficients which are real numbers.

Step by step solution

01

Definition

A polynomial in \(x\) is a mathematical expression that can be written in the following general form: \(p(x) = a_nx^n + a_{n-1}x^{n-1} + ... + a_2x^2 + a_1x + a_0\). Here, \(n\) is a non-negative integer, and \(a_0, a_1, ..., a_n\) are the coefficients which are real numbers. It can also be expressed as a sum of monomials.
02

Breakdown of Components

Looking at the parts of the polynomial, \(x\) is known as the variable or indeterminate, while the \(a\) values are coefficients. The highest power of \(x\) for which the corresponding coefficient is not zero is referred to as the degree of the polynomial. Each term \(a_ix^i\) of the polynomial is called a monomial.
03

Examples

For example, \(p(x) = 5x^4 - 3x^2 + 2x - 7\) is a polynomial in \(x\) of degree 4 with coefficients 5, -3, 2, and -7 for \(x^4, x^2, x, and the constant term, respectively.

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Most popular questions from this chapter

Factor and simplify each algebraic expression. $$-8(4 x+3)^{-2}+10(5 x+1)(4 x+3)^{-1}$$

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I factored \(4 x^{2}-100\) completely and obtained \((2 x+10)(2 x-10)\)

Give an example of a rational number that is not an integer.

List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. $$\left\\{-9,-\frac{4}{5}, 0,0.25, \sqrt{3}, 9.2, \sqrt{100}\right\\}$$

Using an example, explain how to factor out the greatest common factor of a polynomial.
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