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Chapter 3: Problem 77

Synthetic division is a process for dividing a polynomial by \(x-c .\) The coefficient of \(x\) in the divisor is \(1 .\) How might synthetic division be used if you are dividing by \(2 x-4 ?\)

Short Answer

Expert verified
Synthetic division can be used for the division by \(2x - 4\) by first re-expressing it in the form of \(x - c\), which would be \(x - 2\) in this case. Then perform synthetic division as normal using 2 as the synthetic number.

Step by step solution

01

Set Up the Divisor and Dividend

Firstly, express the given divisor, \(2x - 4\), as \(x - 2 \) by dividing through by 2, which makes the coefficient of x as 1. This allows application of the synthetic division. Then, arrange the coefficients of the polynomial (let's denote it as P(x)) to be divided horizontally, aligning degrees of terms.
02

Conduct Synthetic Division

Afterwards, proceed with the synthetic division using 2 (which is the value from the rearranged divisor, \(x - 2\)) as the synthetic number. Synthetic division involves a series of multiplication and addition operations.
03

Interpret the Result

The result of the synthetic division is a set of coefficients which form another polynomial (say Q(x)), which is the quotient. If there is a remainder, then it is written as a fraction over the divisor.

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Key Concepts

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

Polynomial Division
When dealing with polynomials, division is a crucial operation that allows us to simplify expressions or find roots. Unlike regular arithmetic division, dividing polynomials entails separating a dividend, which is a higher-degree polynomial, by a divisor, usually a lower-degree polynomial like a linear or quadratic term. Typically, traditional long division or synthetic division is used for these types of problems.
  • Long Division of Polynomials: Similar to numerical long division, it involves writing the dividend and dividing it step-by-step by the divisor.
  • Synthetic Division: A streamlined method used especially when divisors are of the form \(x-c\), where the coefficient of \(x\) in this context is 1. It is efficient and reduces the complexity of calculations.
Long division can be cumbersome, but synthetic division provides a way to quickly and easily manage these calculations, making it a favorite method for many students when solving polynomial division problems.
Synthetic Substitution
Synthetic substitution is a clever twist on synthetic division. It's often used for evaluating polynomials at certain points, like determining \(P(c)\) where \(x = c\). When you perform synthetic division with \(x - c\), the remainder you get is actually \(P(c)\). This offers a fast way to compute polynomial values without substituting \(x = c\) directly into \(P(x)\).

To use synthetic substitution:
  • Write down the coefficients of the polynomial in descending order of power.
  • Perform synthetic division using \(c\) from the divisor \(x - c\).
  • The final number obtained (the remainder) will be the value of \(P(c)\).
Once you grasp this method, you'll find it not only speeds up your work but also reduces the possibility of errors commonly encountered when substituting directly and calculating manually.
Remainder Theorem
An essential concept related to synthetic division is the Remainder Theorem. This theorem states that when a polynomial \(P(x)\) is divided by \(x-c\), the remainder of this division is \(P(c)\). This is incredibly helpful because it means that by just performing a quick synthetic division, we can find the remainder without fully conducting the long division.

Here's why the Remainder Theorem matters:
  • It provides a shortcut to determine whether \(x-c\) is a factor of \(P(x)\) by checking if \(P(c) = 0\).
  • Using this theorem alongside synthetic division enables fast computation of polynomial remainders.
  • If you know the remainder will be zero, \(x-c\) divides \(P(x)\) evenly, implying \(c\) is a root of the polynomial.
By understanding the Remainder Theorem, you gain another tool in your mathematical toolkit for efficiently handling polynomial equations.

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Most popular questions from this chapter

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{2 x+7}{x+3}$$

a. Use a graphing utility to graph \(y-2 x^{2}-82 x+720\) in a standard viewing rectangle. What do you observe? b. Find the coordinates of the vertex for the given quadratic function. c. The answer to part (b) is \((20.5,-120.5) .\) Because the leading coefficient, \(2,\) of the given function is positive, the vertex is a minimum point on the graph. Use this fact to help find a viewing rectangle that will give a relatively complete picture of the parabola. With an axis of symmetry at \(x=20.5,\) the setting for \(x\) should extend past this, so try \(\mathrm{Xmin}=0\) and \(\mathrm{Xmax}=30 .\) The setting for \(y\) should include (and probably go below) the \(y\) -coordinate of the graph's minimum y-value, so try Ymin \(=-130\). Experiment with Ymax until your utility shows the parabola's major features. d. In general, explain how knowing the coordinates of a parabola's vertex can help determine a reasonable viewing rectangle on a graphing utility for obtaining a complete picture of the parabola.

Describe how to find a parabola's vertex if its equation is expressed in standard form. Give an example.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. As production level increases, the average cost for a company to produce each unit of its product also increases.

What is a polynomial incquality?
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