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Chapter 5: Problem 28

\(\left(14 x^{2}+7 x-21\right) \div(7 x)\)

Short Answer

Expert verified
2x + 1 - \frac{3}{x}

Step by step solution

01

Rewrite the Division Problem

Rewrite the division expression \(\frac{14x^2 + 7x - 21}{7x} \) to make it clear we are dividing each term in the numerator by the denominator.
02

Separate Each Term

Separate the expression into three fractions: \[ \frac{14x^2}{7x} + \frac{7x}{7x} - \frac{21}{7x} \]
03

Simplify Each Term

Simplify each of the three fractions separately: \[ \frac{14x^2}{7x} = 2x \] \[ \frac{7x}{7x} = 1 \] \[ \frac{21}{7x} = 3 / x \] so the simplified expression becomes: \[ 2x + 1 - \frac{3}{x} \]

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Key Concepts

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

simplifying algebraic expressions
To simplify algebraic expressions, you must break down complex expressions into simpler forms. In our exercise, we're dividing a polynomial by a monomial, which involves distributing the division across each term. This means we need to break down the given polynomial and simplify each part separately. Each fraction is simplified by reducing the coefficients and variables individually. For instance, \(\frac{14x^2}{7x}\) requires dividing 14 by 7 and simplifying \x^2 \ by \x\. This results in \2x\. Always remember, the goal is to reduce the expression to its simplest form.
fraction division
Fraction division often seems tricky, but it becomes simpler if you break it down. When dividing fractions, you multiply by the reciprocal. However, in the context of algebra, you distribute the division over all terms in the polynomial. Starting with our example, we distribute the denominator \(7x\) over each term in the numerator: \( \frac{14x^2}{7x} + \frac{7x}{7x} - \frac{21}{7x} \). Here, each term of the numerator is divided by \(7x\). Learning to handle fraction division in this manner builds a strong foundation for more complex operations.
polynomials
Polynomials are algebraic expressions consisting of variables and coefficients, connected by addition, subtraction, and multiplication, but not division by a variable. In our exercise, \(14x^2 + 7x - 21\) is the polynomial being divided by \(7x\). Understanding polynomials helps in breaking them down for operations like addition, subtraction, and division. Here, we break the polynomial into individual fractions where each term of \(14x^2 + 7x - 21\) is divided by \(7x\). This helps us simplify them to \(2x + 1 - \frac{3}{x} \). Remember, polynomials form the building block of many algebraic processes, so grasping this concept is essential.

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Most popular questions from this chapter

\((x+6)(x+6)\)

Problem: Simplify: \((3 p)(p+4)(p+2)\) $$ \text { Incorrect Answer: } \begin{aligned} &(3 p)(p+4)(p+2) \\ &=3 p(p+4)+3 p(p+2) \\ &=3 p(p)+3 p(4)+3 p(p)+3 p(2) \\ &=3 p^{2}+12 p+3 p^{2}+6 p \\ &=6 p^{2}+18 p \end{aligned} $$

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