/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 Perform each division. See Examp... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 3

Perform each division. See Examples 1 through 3. $$ \frac{8 x^{3}-4 x^{2}+6 x+2}{2} $$

Short Answer

Expert verified
The result of the division is \( 4x^3 - 2x^2 + 3x + 1 \).

Step by step solution

01

Set Up the Division

We are given the polynomial \( 8x^3 - 4x^2 + 6x + 2 \) and need to divide it by 2. Think of dividing each term of the polynomial individually by 2.
02

Divide Each Term

Divide each term of the polynomial by 2. Start with the first term: \( \frac{8x^3}{2} = 4x^3 \), then \( \frac{-4x^2}{2} = -2x^2 \), followed by \( \frac{6x}{2} = 3x \), and finally \( \frac{2}{2} = 1 \).
03

Compile the Results

Write the results of each individual division together to form the resulting polynomial: \( 4x^3 - 2x^2 + 3x + 1 \).

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Key Concepts

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

Understanding Polynomials
A polynomial is a mathematical expression consisting of variables and coefficients, structured in terms where each term is composed of a number (coefficient) multiplied by one or more variables raised to a non-negative integer power. For example, in the polynomial \( 8x^3 - 4x^2 + 6x + 2 \), each term includes the variable \( x \) raised to different powers. The highest power of \( x \) in a polynomial determines its degree. In our case, \( 8x^3 \) indicates that the degree is 3.
Polynomials can consist of:
  • Terms: Individual parts of the polynomial, like \( 8x^3 \) or \( 6x \).
  • Coefficients: The numbers in front of the variables, such as 8, -4, and 6.
  • Constants: Terms without variables like the number 2.
Polynomials serve as the building blocks of algebra and are used for modeling various real-world phenomena.
Exploring Algebra
Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. It's the language through which we describe and solve equations and inequalities. At its core, algebra involves finding the unknown values represented by variables.
Key concepts in algebra:
  • Variables: Symbols, often letters like \( x \), used to represent unknown values.
  • Expressions: Combinations of numbers, variables, and operations, like \( 8x^3 - 4x^2 + 6x + 2 \).
  • Equations: Statements of equality between two expressions, often solved to find the value of a variable.
In the given exercise, algebra helps us organize and simplify expressions, making it possible to perform operations like division by isolating terms.
Understanding Mathematical Operations
Mathematical operations are processes that take one or more values and apply a particular process to yield a result. The basic operations include addition, subtraction, multiplication, and division. In the given exercise, the focus is on division, particularly dividing a polynomial by a constant number.
Steps in dividing a polynomial by a constant:
  • Identify each term of the polynomial, such as \( 8x^3 \), \( -4x^2 \), \( 6x \), and \( 2 \).
  • Divide each term individually by the constant, in this case, 2.
  • Compile the results back together to form the final simplified polynomial.
This operation breaks down complex expressions into simpler components. By mastering basic operations in algebra, you can confidently handle more advanced mathematical tasks.

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Most popular questions from this chapter

Subtract. $$ \left(-6 y^{2}+3 y-4\right)-\left(9 y^{2}-3 y\right) $$

Match each expression on the left with its simplification on the right. Not all letters on the right must be used and a letter may be used more than once. a. \(3 y\) b. \(9 y-6 y^{2}\) c. \(10 x\) d. \(25 x^{2}\) e. \(10 x-6\) f. none of these $$ 10 y-6 y^{2}-y $$

Fill in the boxes so that each statement is true. (More than one answer may be possible for these exercises.) $$ \left(x^{\square}\right)^{\square}=x^{-15} $$

Add or subtract as indicated. $$ (14 y+12)+(-3 y-5) $$

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