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Chapter 5: Problem 94

Explain how to divide a polynomial that is not a monomial by a monomial. Give an example.

Short Answer

Expert verified
To divide a polynomial by a monomial, each term of the polynomial should be divided separately by the monomial. For example, the polynomial \(4x^2 + 2x + 8\) divided by the monomial \(2x\) results in \(x + 1 + 4/x\).

Step by step solution

01

- Understanding the terms

A monomial is an algebraic expression consisting of one term. For instance, '5x' or '3y^3' are monomials. A polynomial is a mathematical expression involving a sum of powers in one or more variables (terms) multiplied by coefficients. An example of a polynomial could be '4x^2 + 3x + 2'.
02

- Division process

To divide a polynomial by a monomial, each term of the polynomial needs to be divided individually by the monomial. The coefficient (numerical factor) and the variable (letter) in each term of the polynomial are divided separately.
03

- Demonstrative example

As an example, let's divide the polynomial \(4x^2 + 2x + 8\) by the monomial \(2x\). First, divide the coefficient of each term: \(4x^2/2x=x ; 2x/2x=1 ; 8/2x=4/x\). Hence, the result of the division is \(x + 1 + 4/x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Monomial
In algebra, a **monomial** is a simple, single-term expression. It consists of a coefficient and variables that can be raised to a power (usually a non-negative integer). For example, in the monomial expression \(5x^3\), \(5\) is the coefficient, and \(x^3\) is the variable term.
It's important to highlight that:
  • Monomials have only one term.
  • They can have multiple variables or powers, but all are part of one term.
When dividing by a monomial, you're dealing with a straightforward operation where each part of the polynomial is addressed with the same simplicity of operations that you would apply to a single term. This characteristic makes monomials fundamental in dividing polynomials with ease.
Polynomial
A **polynomial** is an algebraic expression involving multiple terms. These terms are notable for being the sum of variables raised to varying exponents, each multiplied by their respective coefficients. For example, \(3x^2 + 4x + 5\) is a polynomial because it has three terms: each term having a coefficient and a variable.
Key points about polynomials include:
  • They consist of two or more monomials combined.
  • The degree of a polynomial is determined by the highest power of the variable.
  • They can have variables and constants.
Understanding polynomials is critical, as they are pervasive in algebra and offer an expanded view of expressions beyond what simple monomials provide. When performing polynomial division, each term is independently divided, showcasing the polynomial鈥檚 segmented nature.
Algebraic Expression
An **algebraic expression** is a combination of numbers, variables, and operations (like addition, subtraction, multiplication, and division) that represent a value or a formula. Algebraic expressions can include monomials, polynomials, or more complex forms.
Some main features of algebraic expressions include:
  • They can range from simple monomials to intricate polynomials.
  • They involve various operations and mathematical elements.
  • They are essential for constructing equations and functions in algebra.
Algebraic expressions are pivotal in understanding how different terms relate within a mathematical context. They provide the structure upon which polynomial division operates, allowing expressions to be simplified or manipulated effectively.
Coefficient
A **coefficient** in an algebraic expression is the numerical factor preceding a variable. In \(7y^2\), the coefficient is \(7\). Every term within an expression has a coefficient, which indicates the intensity or the scale of that variable term.
Some points to remember include:
  • Coefficients can be positive or negative numbers.
  • In polynomial division, each term's coefficient is divided separately when dividing by a monomial.
  • They play an integral role in determining the expression's overall value.
Coefficients streamline the process of dealing with variables by providing a consistent, scalable measure of their influence in an expression. In operations like polynomial division, managing coefficients is key to accurate simplification and solution formation.

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Most popular questions from this chapter

When dividing a polynomial by a binomial, explain when to stop dividing.

Add or subtract as indicated. $$\left(x^{3}+7 x y-5 y^{2}\right)-\left(6 x^{3}-x y+4 y^{2}\right)$$

Use the FOIL method to find each product. Express the product in descending powers of the variable. $$(6 x-5)(2-x)$$

Use the motion formula \(d=r t,\) distance equals rate times time, and the fact that light travels at the rate of \(1.86 \times 10^{5}\) miles per second, to solve Exercises \(143-144\) If the moon is approximately \(2.325 \times 10^{5}\) miles from Earth, how many seconds does it take moonlight to reach Earth?

Use the FOIL method to find each product. Express the product in descending powers of the variable. $$\left(5 x^{2}-4\right)\left(3 x^{2}-7\right)$$
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