Problem 6
Question
Divide the expression. $$\frac{2 x^{4}-3 x^{2}+4 x-7}{-4 x}$$
Step-by-Step Solution
Verified Answer
The expression is \(-\frac{1}{2}x^3 + \frac{3}{4}x - 1 + \frac{7}{4x}\).
1Step 1: Understand the Problem
We are given a polynomial expression \( \frac{2x^4 - 3x^2 + 4x - 7}{-4x} \) and need to divide each term of the polynomial by \(-4x\).
2Step 2: Divide Each Term Separately
Divide each term of the polynomial by \(-4x\). The expression becomes:\[ \frac{2x^4}{-4x} - \frac{3x^2}{-4x} + \frac{4x}{-4x} - \frac{7}{-4x} \]
3Step 3: Simplify Each Term
Simplify each fraction:- \( \frac{2x^4}{-4x} = -\frac{1}{2}x^3 \) because \( 2x^4 \div -4x = -\frac{1}{2}x^{4-1} \)- \( \frac{-3x^2}{-4x} = \frac{3}{4}x \) as \( -3x^2 \div -4x = \frac{3}{4}x^{2-1} \)- \( \frac{4x}{-4x} = -1 \) since the x cancels out and \( 4 \div -4 = -1 \)- \( \frac{-7}{-4x} = \frac{7}{4x} \) remains as is because \( -7 \div -4x = \frac{7}{4x} \).
4Step 4: Write the Final Expression
Combine the results from the previous step:\[ -\frac{1}{2}x^3 + \frac{3}{4}x - 1 + \frac{7}{4x} \]
Key Concepts
Rational ExpressionsPolynomial SimplificationAlgebraic Fractions
Rational Expressions
Rational expressions are similar to fractions, but instead of being made up of integers, they involve polynomials. A rational expression is essentially a ratio of two polynomials, with one polynomial in the numerator and another in the denominator.
This concept arises often in algebra and calculus, serving as a foundation for understanding more complex mathematical topics.
For example, in the expression \(\frac{2x^4 - 3x^2 + 4x - 7}{-4x}\), the numerator is the polynomial \(2x^4 - 3x^2 + 4x - 7\) and the denominator is another polynomial, \(-4x\). Rational expressions can be simplified and manipulated similarly to numerical fractions, allowing for operations like addition, subtraction, and division to be performed between them.
This concept arises often in algebra and calculus, serving as a foundation for understanding more complex mathematical topics.
For example, in the expression \(\frac{2x^4 - 3x^2 + 4x - 7}{-4x}\), the numerator is the polynomial \(2x^4 - 3x^2 + 4x - 7\) and the denominator is another polynomial, \(-4x\). Rational expressions can be simplified and manipulated similarly to numerical fractions, allowing for operations like addition, subtraction, and division to be performed between them.
- Recognize that a rational expression looks like a fraction with polynomials.
- Understand that the same rules for fractions apply to rational expressions, such as the need for a common denominator in addition.
- Factorization often helps in simplifying rational expressions.
Polynomial Simplification
Polynomial simplification involves reducing a polynomial to its simplest form. This means eliminating any unnecessary terms and combining like terms where possible.
Simplification can make complex expressions easier to work with and understand.
When dealing with polynomial division, as in \(\frac{2x^4 - 3x^2 + 4x - 7}{-4x}\), it's important to divide each term of the numerator separately by the term in the denominator. This allows each part to be simplified independently, transforming what might initially seem like a convoluted expression into something more manageable.
Simplification can make complex expressions easier to work with and understand.
When dealing with polynomial division, as in \(\frac{2x^4 - 3x^2 + 4x - 7}{-4x}\), it's important to divide each term of the numerator separately by the term in the denominator. This allows each part to be simplified independently, transforming what might initially seem like a convoluted expression into something more manageable.
- Simplifying \(\frac{2x^4}{-4x}\) yields \(-\frac{1}{2}x^3\), reducing the degree of the numerator by one.
- In \(\frac{-3x^2}{-4x}\), similar simplification leads to \(\frac{3}{4}x\).
- Understanding how to simplify each division separately ensures a correct and complete simplification of the polynomial.
Algebraic Fractions
Algebraic fractions involve variables in the numerator, denominator, or both, and function similarly to rational expressions. They are fractions that have algebraic expressions on the top, bottom, or both.
Working with algebraic fractions requires knowledge of both fraction operations and algebraic manipulation skills.
For instance, in the expression \(\frac{2x^4 - 3x^2 + 4x - 7}{-4x}\), the algebraic fraction highlights how algebraic expressions can be presented and manipulated in fraction form. This includes performing operations like simplification by canceling out common terms when possible and transforming complex fractions into simpler forms.
Working with algebraic fractions requires knowledge of both fraction operations and algebraic manipulation skills.
For instance, in the expression \(\frac{2x^4 - 3x^2 + 4x - 7}{-4x}\), the algebraic fraction highlights how algebraic expressions can be presented and manipulated in fraction form. This includes performing operations like simplification by canceling out common terms when possible and transforming complex fractions into simpler forms.
- Learn to perform basic fraction operations with algebraic terms like addition and multiplication.
- Practice transforming complex algebraic fractions into simpler forms to make calculations easier.
- Recognize the importance of maintaining the integrity of the denominator, ensuring that expressions are not undefined by division by zero.
Other exercises in this chapter
Problem 5
Determine whether \(f\) is a rational function and state its domain. $$ f(x)=\frac{|x-1|}{x+1} $$
View solution Problem 5
Determine if the junction is a polynomial function. If it is, state its degree and leading coefficient a. $$ f(x)=1-4 x-5 x^{4} $$
View solution Problem 6
Evaluate the expression by hand. $$ 32^{1 / 5} $$
View solution Problem 6
Solve the rational equation (a) symbolically, (b) graphically, and (c) numerically $$ \frac{4}{x-2}=\frac{3}{x-1} $$
View solution