Dividing Polynomial Expressions A Step-by-Step Guide

Jul 7, 2025 by Admin 53 views

To solve the division problem $\frac{15 b^9-35 b^7+35 b^4}{5 b^4}$ , we need to divide each term in the numerator by the denominator $5b^4$ . This involves applying the distributive property of division over addition and subtraction. We will simplify each term individually and then combine the results.

Step-by-step Solution

Divide the first term:

The first term in the numerator is $15b^9$ . We divide this by $5b^4$ :
$\frac{15 b^9}{5 b^4} = \frac{15}{5} \cdot \frac{b^9}{b^4}$
We simplify the coefficients and use the quotient rule for exponents, which states that $\frac{b^m}{b^n} = b^{m-n}$ :
$\frac{15}{5} \cdot b^{9-4} = 3 b^5$
Divide the second term:

The second term in the numerator is $-35b^7$ . We divide this by $5b^4$ :
$\frac{-35 b^7}{5 b^4} = \frac{-35}{5} \cdot \frac{b^7}{b^4}$
Simplify the coefficients and apply the quotient rule for exponents:
$-7 \cdot b^{7-4} = -7 b^3$
Divide the third term:

The third term in the numerator is $35b^4$ . We divide this by $5b^4$ :
$\frac{35 b^4}{5 b^4} = \frac{35}{5} \cdot \frac{b^4}{b^4}$
Simplify the coefficients and apply the quotient rule for exponents:
$7 \cdot b^{4-4} = 7 \cdot b^0$
Since any non-zero number raised to the power of 0 is 1, we have:
$7 \cdot 1 = 7$
Combine the results:

Now, we combine the simplified terms:
$3b^5 - 7b^3 + 7$

So, the simplified expression is $3b^5 - 7b^3 + 7$ .

Detailed Explanation of Each Step

Dividing the First Term: $15b^9$ by $5b^4$

When dividing $15b^9$ by $5b^4$ , we first focus on the coefficients. The coefficient of the first term is 15, and the coefficient of the divisor is 5. Dividing 15 by 5 yields 3. Next, we consider the variable part, which is $b^9$ divided by $b^4$ . According to the quotient rule of exponents, when dividing like bases, we subtract the exponents. Therefore, $\frac{b^9}{b^4} = b^{9-4} = b^5$ . Combining these results, we find that $\frac{15b^9}{5b^4} = 3b^5$ .

Dividing the Second Term: $-35b^7$ by $5b^4$

For the second term, we are dividing $-35b^7$ by $5b^4$ . Again, we start by dividing the coefficients. The coefficient of the second term is -35, and the coefficient of the divisor is 5. Dividing -35 by 5 gives us -7. For the variable part, we have $b^7$ divided by $b^4$ . Applying the quotient rule of exponents, we get $\frac{b^7}{b^4} = b^{7-4} = b^3$ . Thus, $\frac{-35b^7}{5b^4} = -7b^3$ .

Dividing the Third Term: $35b^4$ by $5b^4$

Finally, we divide $35b^4$ by $5b^4$ . The coefficients are 35 and 5. Dividing 35 by 5 results in 7. For the variable part, we have $b^4$ divided by $b^4$ . Using the quotient rule of exponents, we have $\frac{b^4}{b^4} = b^{4-4} = b^0$ . By definition, any non-zero number raised to the power of 0 is 1. Therefore, $b^0 = 1$ , and $\frac{35b^4}{5b^4} = 7 \cdot 1 = 7$ .

Combining the Simplified Terms

Now that we have simplified each term individually, we combine the results. We found that $\frac{15b^9}{5b^4} = 3b^5$ , $\frac{-35b^7}{5b^4} = -7b^3$ , and $\frac{35b^4}{5b^4} = 7$ . Adding these together, we get the final simplified expression:

3b^5 - 7b^3 + 7

Importance of Understanding Polynomial Division

Understanding polynomial division is crucial in various areas of mathematics, including algebra and calculus. It is a fundamental skill that enables you to simplify complex expressions, solve equations, and perform advanced mathematical operations. Polynomial division is not just an isolated concept; it is interconnected with many other mathematical principles and applications.

Applications in Algebra

In algebra, polynomial division is used to factor polynomials, find roots, and simplify rational expressions. When dealing with higher-degree polynomials, division helps break them down into simpler factors, making it easier to find solutions to polynomial equations. Factoring polynomials is a cornerstone of algebraic manipulation and is essential for solving various problems.

Applications in Calculus

In calculus, polynomial division is often used to simplify rational functions before integration or differentiation. Rational functions, which are ratios of two polynomials, can sometimes be challenging to work with directly. By performing polynomial division, we can rewrite these functions into a more manageable form, often making integration or differentiation straightforward. This technique is particularly useful when dealing with improper rational functions, where the degree of the numerator is greater than or equal to the degree of the denominator.

Real-World Applications

Beyond academic applications, polynomial division has practical uses in various real-world scenarios. For instance, in engineering, it can be used to model and solve problems related to signal processing and control systems. In computer science, polynomial division is used in coding theory and cryptography. The ability to manipulate and simplify polynomial expressions is a valuable skill in these fields.

Common Mistakes to Avoid

When performing polynomial division, several common mistakes can occur. Being aware of these pitfalls can help you avoid them and ensure accurate results.

Forgetting to Distribute

A frequent mistake is forgetting to distribute the division across all terms in the numerator. Each term must be divided by the denominator. For example, in the given problem, each of the terms $15b^9$ , $-35b^7$ , and $35b^4$ must be divided by $5b^4$ . Neglecting to do this for even one term will lead to an incorrect answer.

Incorrectly Applying the Quotient Rule

The quotient rule of exponents, which states that $\frac{b^m}{b^n} = b^{m-n}$ , is crucial in polynomial division. A common error is either adding the exponents instead of subtracting them or miscalculating the difference. Always double-check that you are correctly applying this rule.

Errors with Coefficients

Coefficient errors are also common. Make sure to correctly divide the numerical coefficients. Simple arithmetic mistakes, such as dividing 15 by 5 and getting 2 instead of 3, can easily occur if not careful. It’s always a good practice to review your arithmetic calculations.

Misunderstanding $b^0$

Another mistake involves the term $b^0$ . Remember that any non-zero number raised to the power of 0 is 1. Failing to recognize this can lead to errors in the final simplification.

Not Simplifying Completely

Finally, ensure that you simplify the expression completely. Combine like terms and make sure there are no further simplifications possible. Leaving an expression partially simplified is not only incorrect but also shows a lack of understanding of the process.

Conclusion

The solution to the division problem $\frac{15 b^9-35 b^7+35 b^4}{5 b^4}$ is $3b^5 - 7b^3 + 7$ . This result is obtained by dividing each term in the numerator by the denominator and applying the quotient rule for exponents. Polynomial division is a vital skill in mathematics with various applications in algebra, calculus, and real-world problems. Avoiding common mistakes and understanding the underlying principles will help you master this concept.