/*! 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 33 Perform each division. $$ \l... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 33

Perform each division. $$ \left(20 a^{4}-15 a^{5}+25 a^{3}\right) \div\left(5 a^{4}\right) $$

Short Answer

Expert verified
4 - 3a + 5a^{-1}

Step by step solution

01

Rewrite the division as fractions

Rewrite the given division problem by expressing each term in the dividend as a fraction over the divisor. \[\frac{20a^4}{5a^4} - \frac{15a^5}{5a^4} + \frac{25a^3}{5a^4}\]
02

Simplify each fraction

Simplify each term by dividing the coefficients and subtracting the exponents of the same bases. \[\frac{20a^4}{5a^4} = 4, \quad \frac{15a^5}{5a^4} = 3a, \quad \frac{25a^3}{5a^4} = 5a^{-1}\]
03

Combine the simplified terms

Write the simplified terms together to get the final simplified expression. \[4 - 3a + 5a^{-1}\]

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Key Concepts

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

Polynomial Division
Polynomial division can seem tricky at first, but it becomes easier with practice. It involves dividing two polynomials to get a quotient and sometimes a remainder. Think of it like long division but with polynomials. For example, if you have \(20a^4 - 15a^5 + 25a^3\) divided by \(5a^4\), you can break it down term by term. The entire polynomial gets divided by \(5a^4\), making it easier to handle. Take each term of the dividend and divide it by the divisor separately.
Simplifying Fractions
Simplifying fractions is essential to make calculations easier. In polynomial division, like in our step-by-step solution, you need to simplify each term. Each fraction \(\frac{numerator}{denominator}\) should be broken down by simplifying the coefficients (the numbers in front of the variables) and reducing the variables using properties of exponents. For instance, \(\frac{20a^4}{5a^4}\) simplifies to \4\ by dividing 20 by 5 and canceling out the \(a^4\). Always remember: simplify coefficients and reduce powers!
Exponents Subtraction
When you divide terms with the same base, you subtract the exponents. This is a fundamental rule in algebra. For example, in \(\frac{15a^5}{5a^4}\), you subtract the exponent 4 from 5, leaving you with \(a^{1}\) or just \a\. Since \5\ gets divided by \5\, the terms simplify to \3a\. Another tricky part is dealing with negative exponents. For \(a^{-1}\), it indicates a reciprocal. So \(5a^{-1}\) is the same as \(\frac{5}{a}\). Remember: subtract exponents when dividing similar bases and interpret negative exponents as reciprocals.

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Most popular questions from this chapter

Find each product. In Exercises \(81-84,89,\) and \(90,\) apply the meaning of exponents. $$ 3 p^{3}\left(2 p^{2}+5 p\right)\left(p^{3}+2 p+1\right) $$

Perform each division. $$ \frac{-4 x+3 x^{3}+2}{x-1} $$

Evaluate. $$ 49 \div 10,000 $$

Find each product. \(-3 k\left(8 k^{2}-12 k+2\right)\)

Perform each division using the "long division" process. $$ \frac{2 r^{3}-5 r^{2}-6 r+15}{r-3} $$
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