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Chapter 5: Problem 127

Factor out the specified factor. $$ t^{-3} \text { from } t^{5}+4 t^{-6} $$

Short Answer

Expert verified
The expression is factored as \( t^{-3}(t^8 + 4t^{-3}) \).

Step by step solution

01

Identify the common factor

Notice that both terms in the expression contain powers of \( t \). The problem instructs us to factor out \( t^{-3} \). This means that \( t^{-3} \) is our common factor or the term to be factored out of the expression \( t^5 + 4t^{-6} \).
02

Divide each term by the common factor

Divide each term by \( t^{-3} \):1. For the first term \( t^5 \), divide it by \( t^{-3} \):\[\frac{t^5}{t^{-3}} = t^{5 - (-3)} = t^8\]2. For the second term \( 4t^{-6} \), divide it by \( t^{-3} \):\[\frac{4t^{-6}}{t^{-3}} = 4t^{-6 - (-3)} = 4t^{-3}\]
03

Write the factored expression

Now, the expression can be rewritten using the common factor \( t^{-3} \). The factored form of the expression is:\[t^{-3}(t^8 + 4t^{-3})\]

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Key Concepts

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

Understanding the Common Factor
When we talk about factoring expressions, especially polynomial expressions, we're often referring to the process of identifying and extracting a common factor. A common factor is a term that divides each part of the expression evenly.
In the example provided, the common factor was specified as \( t^{-3} \).
Our task was to "factor out" this common factor from the expression \( t^5 + 4t^{-6} \).

This process involves two steps:
  • First, recognizing that both terms (\( t^5 \) and \( 4t^{-6} \)) contain the variable \( t \) raised to a power.
  • Second, confirming that \( t^{-3} \) can be divided out from each term.
Once identified, we use division to separate the common factor from the rest of the expression, simplifying the original equation into a more workable form.
Playing with Exponents
Exponents are a way of expressing repeated multiplication of a number by itself. They are crucial in simplifying expressions and equations, including factoring.
In the original exercise, we see exponents used to describe powers of \( t \).

Here's a quick refresher on handling exponents:
  • When dividing with the same base, you subtract the exponents: \( \frac{a^m}{a^n} = a^{m-n} \).
  • This rule is critical when factoring expressions like our example \( t^5 + 4t^{-6} \).
  • When dealing with negative exponents, remember that \( a^{-n} = \frac{1}{a^n} \).
In our exercise, understanding these rules allowed us to convert and simplify exponents correctly, making it possible to factor the expression down efficiently.
Unpacking Polynomial Expressions
Polynomial expressions are mathematical phrases that involve sums of powers of variables.
In our example, we're dealing with the polynomial expression \( t^5 + 4t^{-6} \).
Polynomials can have constants, variables, and exponents, but the key idea is to organize and simplify them through various techniques, one of which is factoring out common factors.

The steps we followed:
  • Identifying the common factors in each term.
  • Utilizing exponent rules to divide each term by the common factor.
  • Rewriting the expression in its factored form, simplifying a previously daunting expression into something manageable.
Being comfortable with polynomials and their manipulation through factoring is essential for handling a wide variety of mathematical tasks effectively. They're everywhere in math, so mastering them opens the door to better mathematical understanding and problem-solving.

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Most popular questions from this chapter

Factor the expression in part a. Then use your answer from part a to give the factorization of the expression in part b. (No new work is necessary!) a. \(c^{3}-\frac{1}{8}\) b. \(c^{3}+\frac{1}{8}\)

To make boxes, a manufacturer cuts equal-sized squares from each corner of the 10 in. \(\times 12\) in. piece of cardboard shown below and then folds up the sides. The polynomial function \(f(x)=4 x^{3}-44 x^{2}+120 x\) gives the volume (in cubic inches) of the resulting box when a square with sides \(x\) inches long is cut from each corner. Find the volume of a box if 3 -inch squares are cut out. (PICUTURE NOT COPY)

Factor each expression. $$ r^{12}+s^{18} r^{24} $$

Find a quadratic equation with solutions \(\frac{1}{4}\) and \(-\frac{4}{3}\)

Multiply. Assume \(n\) is a natural number. $$ a^{2 n}\left(a^{n}+a^{2 n}\right) $$
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