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Chapter 6: Problem 14

Divide and check. $$ \left(6 a^{4}+9 a^{2}-8\right) \div(2 a) $$

Short Answer

Expert verified
3a^3 + \frac{9}{2}a - 4a^{-1}

Step by step solution

01

Rewrite the Division

Rewrite the given expression by distributing the division to each term in the numerator separately: \[ (6a^4 + 9a^2 - 8) \div (2a) \Rightarrow \frac{6a^4}{2a} + \frac{9a^2}{2a} - \frac{8}{2a} \]
02

Simplify Each Term

Divide each term separately within the fraction: \[ \frac{6a^4}{2a} = 3a^3 \]\[ \frac{9a^2}{2a} = \frac{9}{2} a \]\[ \frac{8}{2a} = 4a^{-1} \]
03

Combine the Simplified Terms

Combine the simplified terms together:\[ 3a^3 + \frac{9}{2}a - 4a^{-1} \]
04

Verify the Result

Multiply the result by the divisor to check the division. The original expression should be obtained if the division was correct.

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Key Concepts

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

Rational Expressions
Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding them helps solve various algebraic problems including division.
When working with rational expressions, you often need to simplify, add, subtract, multiply, or divide them.
Here鈥檚 how you work with rational expressions:
  • Factor both the numerator and the denominator when possible.
  • Look for and cancel out common factors.
  • Simplify the fractions by performing the indicated operations.
In our exercise, the rational expression is \((6a^4 + 9a^2 - 8) 梅 (2a)\). By distributing the divisor \(2a\) across each term in the numerator, we simplify each term into a manageable form.
Simplifying Fractions
Simplifying fractions involves reducing them to their simplest form. This helps in making calculations easier and understanding the relations between different terms.
To simplify fractions, follow these steps:
  • Divide the numerator and the denominator by their greatest common factor (GCF).
  • Rewrite the fraction with these simplified parts.
In our exercise, we apply this principle by dividing each term of the expression individually by \(2a\) to get simplifying fractions:
\(\frac{6a^4}{2a}\) simplifies to \(3a^3\), \(\frac{9a^2}{2a}\) as \(\frac{9}{2}a\), and \(\frac{8}{2a}\) to \(\frac{4}{a}\). This breaks a complex polynomial into simpler fractions, making it easier to work with.
Exponent Rules
Exponent rules are fundamental when dealing with polynomials, especially during division. Understanding these rules makes simplifying expressions straight-forward:
Important exponent rules include:
  • Product of Powers: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power: \((a^m)^n = a^{m \times n}\)
  • Quotient of Powers: \(\frac{a^m}{a^n} = a^{m-n}\)
In this exercise, we use the quotient of powers rule to simplify individual fractions:
\(\frac{6a^4}{2a}\) simplifies to \(3a^3\) because \(a^{4-1} = a^3\). The same principle is applied to other terms. Becoming familiar with these rules is crucial for handling algebraic expressions efficiently.

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Most popular questions from this chapter

Explain the difference between adding rational expressions and solving rational equations.

For each pair of functions fand \(g,\) find all values of a for which \(f(a)=g(a)\) $$ \begin{array}{l}{f(x)=\frac{2 x+5}{x^{2}+4 x+3}} \\\ {g(x)=\frac{x+2}{x^{2}-9}+\frac{x-1}{x^{2}-2 x-3}}\end{array} $$

Find and simplify $$ \frac{f(x+h)-f(x)}{h} $$ for each rational function \(f\) $$ f(x)=\frac{3}{x} $$

For each pair of functions fand \(g,\) find all values of a for which \(f(a)=g(a)\) $$ \begin{array}{l}{f(x)=\frac{4}{x^{2}+3 x-10}} \\\ {g(x)=\frac{3}{x^{2}-x-12}+\frac{1}{x^{2}+x-6}}\end{array} $$

Simplify. $$ \frac{x^{5}-x^{3}+x^{2}-1-\left(x^{3}-1\right)(x+1)^{2}}{\left(x^{2}-1\right)^{2}} $$
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