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Chapter 8: Problem 88

Factor expression. \(56 r s^{2} t^{3}+7 r s^{2} v^{6}\)

Short Answer

Expert verified
The expression factors to \(7rs^2(8t^3 + v^6)\).

Step by step solution

01

Identify the Greatest Common Factor (GCF)

First, look for the largest factor that is common to both terms in the expression. In this case, both terms have the variable components \(r\) and \(s^2\). The coefficients are 56 and 7, where 7 is a common factor.
02

Extract the GCF from the Expression

The GCF identified is \(7rs^2\). Factor this out from the expression: \( 56rs^2t^3 + 7rs^2v^6 = 7rs^2(8t^3 + v^6) \). Here, 56 divided by 7 gives 8, and the remaining terms are \(t^3\) in the first term and \(v^6\) in the second term.

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Key Concepts

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

Greatest Common Factor
When dealing with algebraic expressions, identifying the Greatest Common Factor (GCF) is a key first step in factoring. The GCF is the largest factor that is shared by all the terms within an expression. To find it, you look at both the numerical coefficients and any variable factors that are present in each term.
  • Start by looking at the numerical part, which are the coefficients. For example, in the expression \(56 r s^{2} t^{3} + 7 r s^{2} v^{6}\), the coefficients are 56 and 7. The greatest common factor of 56 and 7 is 7, because 7 is the largest number that divides evenly into both.
  • Next, examine the variable components. Common variables such as \(r\) and \(s^2\) appear in both terms, making them part of the GCF.
By systematically identifying the common numerical and variable components, you determine the GCF for the expression, which in this case is \(7rs^2\). Factoring involves reconstructing the original expression by extracting this greatest common factor from each term.
Variable Components
Variable components in an expression are simply the letters that represent numbers. They can have different powers and are usually involved in expressions where simplification or factoring is needed. In our example expression, \(56 r s^{2} t^{3} + 7 r s^{2} v^{6}\), the variable components are \(r\), \(s^2\), \(t^3\), and \(v^6\).
  • Diving deeper, \(r\) and \(s^2\) are repeated in both terms. This means they are shared across the expression, and they'll be part of our GCF.
  • While \(t^3\) is unique to the first term and \(v^6\) is unique to the second term, these variables do not form part of the GCF because they are not common to both terms.
The key is to identify variable parts that are consistent across terms, as they will help simplify and factor the expression correctly.
Coefficients
Coefficients are the numeric parts of terms in algebraic expressions. They show how many times a term is multiplied and are a fundamental part of determining the GCF of an expression. In our specific example, the expression \(56 r s^{2} t^{3} + 7 r s^{2} v^{6}\) contains the coefficients 56 and 7.
  • The coefficient 56 indicates that the term is 56 times its variable counterpart, while 7 indicates the second term has the variables multiplied by 7.
  • Finding the GCF of these coefficients was straightforward, as 7 is the greatest number that evenly divides both 56 and 7.
Once you find this numerical GCF, it becomes part of the factorization process. This means you can reduce or factor nearly any algebraic expression systematically by identifying and factoring out the common values of the coefficients. This skill is essential for simplifying complex algebraic expressions.

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Most popular questions from this chapter

Roller Coasters. The polynomial function \(f(x)=0.001 x^{3}-0.12 x^{2}+3.6 x+10\) models the path of a portion of the track of a roller coaster. Use the function equation to find the height of the track for \(x=0,20,40,\) and 60.

Concessionaires. A baseball club pays a vendor 125 dollars per game for selling bags of peanuts for 4.75 dollars each. a. Write a linear function that describes the profit the vendor makes for the baseball club during a game if she sells \(b\) bags of peanuts. b. Find the profit the baseball club will make if the vendor sells 110 bags of peanuts during a game.

Breathing Capacity. When fitness instructors prescribe exercise workouts for elderly patients, they must take into account age-related loss of lung function. Studies show that the percent of remaining breathing capacity for someone over 30 years old can be modeled by a linear function. (Source: alsearsmd.com) a. At 35 years of age, approximately \(90 \%\) of maximal breathing capacity remains and at 55 years of age, approximately \(66 \%\) of maximal breathing capacity remains. Let \(a\) be the age of a patient and \(L\) be the percent of her maximal breathing capacity that remains. Write a linear function \(L(a)\) to model this situation. b. Use your answer to part a to estimate the percent of maximal breathing capacity that remains in an 80 -year-old.

Write an equation for a linear function whose graph has the given characteristics. See Example 7. Passes through \((1,7)\) and \((-2,1)\)

Perform the operations and simplify, if possible. See Example 5 $$\frac{2 p^{2}-5 p-3}{p^{2}-9} \cdot \frac{2 p^{2}+5 p-3}{2 p^{2}+5 p+2}$$
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