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Chapter 6: Problem 23

Use synthetic division to divide. $$ \frac{x^{2}+5 x-6}{x+6} $$

Short Answer

Expert verified
The quotient is \(x - 1\).

Step by step solution

01

Set up the synthetic division

To use synthetic division, first write down the coefficients of the polynomial \(x^2 + 5x - 6\), which are \([1, 5, -6]\). The divisor \(x+6\) gives us a root \(r = -6\) to use in synthetic division.
02

Perform synthetic division

Write \(-6\) outside the division array. Start by bringing down the leading coefficient \(1\) to the bottom row. Multiply \(-6\) by \(1\) to get \(-6\), and write this under the next coefficient \(5\). Add \(5\) and \(-6\) to get \(-1\), and write it in the bottom row. Repeat this: multiply \(-6\) by \(-1\) to get \(6\), write \(6\) under \(-6\), add to get \(0\). The remainder is \(0\).
03

Write the quotient

The bottom row gives the quotient from the division. The coefficients \([1, -1]\) correspond to the polynomial \(x - 1\). Thus, \(x^2 + 5x - 6\) divided by \(x+6\) gives a quotient of \(x - 1\) with a remainder of \(0\).

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Key Concepts

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

Polynomial Division
Polynomial division is an essential mathematical technique that simplifies expressions involving polynomials. Instead of using traditional long division, one can use synthetic division for polynomials, which is much simpler when the divisor is in the form \( x - r \). Here's what you need to know:
  • Just like number division, polynomial division separates a dividend by a divisor, resulting in a quotient and sometimes a remainder.
  • The goal is to simplify a polynomial expression into a more readable form.
To effectively perform polynomial division, grasp these steps:
  • Identify the polynomial you want to divide, called the dividend.
  • Find a divisor, which in simple cases is often in the format \( x - r \).
  • Utilize a method like synthetic division to conduct the division quickly.
  • Interpret the result as a quotient (a simpler polynomial) and possibly a remainder.
By mastering polynomial division, you can easily manage and interpret data expressed in polynomial forms.
Remainder Theorem
The remainder theorem is a helpful concept in algebra that connects polynomial division and the evaluation of polynomials. Here's how it works:
  • If a polynomial \( f(x) \) is divided by \( x - r \), the remainder of this division is \( f(r) \).
  • This means you can determine the remainder simply by evaluating the polynomial at the given root \( r \).
Using the remainder theorem helps in:
  • Quickly checking if a number \( r \) is a root of the polynomial. A zero remainder indicates that \( r \) is indeed a root.
  • Simplifying computations while handling polynomial division.
In practice, the remainder theorem makes it easier to factor polynomials and determine their behavior at specific points without performing the complete division.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can include numbers, variables, and operators. Understanding these is vital to handling more complex math problems.Key components include:
  • **Variables**: Symbols that represent unknown or variable quantities, commonly denoted by letters like \( x, y, \) or \( z \).
  • **Operators**: Symbols that show operations, such as \(+, -, *, /\).
  • **Terms**: The building blocks of expressions, consisting of numbers and variables connected by multiplication or division.
Understanding algebraic expressions allows:
  • Simplification and manipulation of mathematical problems.
  • Solutions to polynomial equations through substitution and division techniques.
  • Clear communication and representation of mathematical ideas.
Grasping algebraic expressions prepares you for tackling more sophisticated operations like polynomial division using synthetic division.

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Most popular questions from this chapter

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Kraft Foods is a provider of many of the best-known food brands in our supermarkets. Among their wellknown brands are Kraft, Oscar Mayer, Maxwell House, and Oreo. Kraft Foods' annual revenues since 2005 can be modeled by the polynomial function \(R(x)=0.06 x^{3}+0.02 x^{2}+1.67 x+32.33,\) where \(R(x)\) is revenue in billions of dollars and \(x\) is the number of years since \(2005 .\) Kraft Foods' net profit can be modeled by the function \(P(x)=0.07 x^{3}-0.42 x^{2}+0.7 x+2.63,\) where \(P(x)\) is the net profit in billions of dollars and \(x\) is the number of years since \(2005 .\) (Source: Based on information from Kraft Foods) a. Suppose that a market analyst has found the model \(P(x)\) and another analyst at the same firm has found the model \(R(x) .\) The analysts have been asked by their manager to work together to find a model for Kraft Foods' profit margin. The analysts know that a company's profit margin is the ratio of its profit to its revenue. Describe how these two analysts could collaborate to find a function \(m(x)\) that models Kraft Foods' net profit margin based on the work they have done independently. b. Without actually finding \(m(x),\) give a general description of what you would expect the answer to be.
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