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Chapter 5: Problem 18

Perform each division. $$ \frac{-13 t^{9}+8 t^{6}-t^{5}}{-t^{6}} $$

Short Answer

Expert verified
13 t^{3} - 8 + t^{-1}

Step by step solution

01

Recognize the Division Format

Identify that each term in the numerator \( -13 t^{9} + 8 t^{6} - t^{5} \) needs to be divided by the denominator \( -t^{6} \).
02

Divide Each Term Individually

Divide each term of the numerator by the denominator: \(\frac{-13 t^{9}}{-t^{6}} + \frac{8 t^{6}}{-t^{6}} - \frac{-t^{5}}{-t^{6}}\).
03

Simplify Each Division

Simplify each term: \(-13 t^{9} / -t^{6} = 13 t^{3} \, \ 8 t^{6} / -t^{6} = -8 \, and \ -t^{5} / -t^{6} = \frac{1}{t} \).
04

Combine the Simplified Terms

Combine the simplified results: \( 13 t^{3} - 8 + t^{-1} \). Note that \( t^{-1} \) is the same as \ \frac{1}{t} \.

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Key Concepts

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

Understanding Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations such as addition, subtraction, multiplication, and division. In our problem, we have an expression \-13 t^{9} + 8 t^{6} - t^{5}\, which includes variables (\(t\)), coefficients (-13, 8, -1), and exponents (9, 6, 5). Interpreting these expressions correctly is critical for solving polynomial division problems. Each term is separated by a plus or minus sign. Knowing this allows you to perform division term-by-term.
Term-by-Term Division
When dividing polynomials, it's essential to divide each term in the numerator by the denominator individually. This method simplifies the problem and makes it more manageable. In our exercise, we need to divide each term in \(-13 t^{9} + 8 t^{6} - t^{5}\) by \(-t^{6}\). This operation is akin to splitting the original problem into smaller, simpler fractions:
\frac{-13 t^{9}}{-t^{6}} + \frac{8 t^{6}}{-t^{6}} - \frac{-t^{5}}{-t^{6}}\. You'll handle each division separately before combining the results.
Simplifying Expressions
Once you divide each term, you need to simplify the results. This involves applying basic arithmetic and the rules of exponents. For instance, \(-13 t^{9} / -t^{6}\) simplifies to \(13 t^{3}\) because \(-13 / -1 = 13\) and subtracting the exponents \(9 - 6 = 3\). Similarly, simplifying \(8 t^{6} / -t^{6}\) yields \(-8\) and \(- t^{5} / -t^{6}\) becomes \(1/t\) or \(t^{-1}\). Combining these results, we get \(13 t^{3} - 8 + t^{-1}\).
Exponent Rules
Exponents express repeated multiplication of a base. When dividing terms with the same base, you subtract the exponents. For example, \(\frac{a^{m}}{a^{n}} = a^{m-n}\). In our problem, each term in the numerator has \(t\) as the base and exponents (9, 6, 5), which we divide by \(-t^{6}\). Using the rule, \(t^{9} / t^{6}\) results in \(t^{9-6} = t^{3}\). This method applies to all terms, resulting in simplified exponents in the output expression.

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Most popular questions from this chapter

Evaluate. $$ 6504 \div 100 $$

Use scientific notation to calculate the answer to each problem. During the \(2007-2008\) season, Broadway shows grossed a total of \(9.38 \times 10^{8}\) dollars. Total attendance for the season was \(1.23 \times 10^{7} .\) What was the average ticket price for a Broadway show?

Evaluate. $$ 277 \div 1000 $$

The special product $$ (x+y)(x-y)=x^{2}-y^{2} $$ $$ \text { can be used to perform some multiplication problems. Here are two examples.} $$ $$ \begin{aligned} 51 \times 49 &=(50+1)(50-1) \\ &=50^{2}-1^{2} \\ &=2500-1^{2} \\ &=2499 \end{aligned} \quad | \begin{aligned} 102 \times 98 &=(100+2)(100-2) \\ &=100^{2}-2^{2} \\ &=10,000-4 \\ &=9996 \end{aligned} $$ Once these patterns are recognized, multiplications of this type can be done mentally. Use this method to calculate each product mentally. $$ 101 \times 99 $$

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