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Chapter 4: Problem 48

\(39-51\) . Use synthetic division and the Remainder Theorem to evaluate \(P(c) .\) $$ P(x)=-2 x^{6}+7 x^{5}+40 x^{4}-7 x^{2}+10 x+112, \quad c=-3 $$

Short Answer

Expert verified
\(P(-3) = 100\) after synthetic division.

Step by step solution

01

Set up Synthetic Division

The first step in using synthetic division is to write down the coefficients of the polynomial \(P(x)\). Here, \(P(x) = -2x^6 + 7x^5 + 40x^4 + 0x^3 - 7x^2 + 10x + 112\), so the coefficients are \([-2, 7, 40, 0, -7, 10, 112]\). For synthetic division with \(c = -3\), write \(-3\) to the left and draw a horizontal line to separate the synthetic division workspace.
02

Begin Synthetic Division

Copy the leading coefficient down below the line: \(-2\). Multiply this number by \(-3\) (the divisor) and write the result under the next coefficient: \(-2 \times -3 = 6\). Add this result to the next coefficient above the line (\(7\)): \(7 + 6 = 13\). Write \(13\) below the line.
03

Continue Synthetic Division

Repeat the multiply-and-add process for each column. Multiply \(13\) by \(-3\) and add to the next coefficient: \(13 \times -3 = -39\), \,\(40 - 39 = 1\). Multiply \(1\) by \(-3\) and add to next: \(1 \times -3 = -3\), \, \(0 - 3 = -3\). Continue this process: \(-3 \times -3 = 9\), \, \(-7 + 9 = 2\). Multiply and add: \(2 \times -3 = -6\), \, \(10 - 6 = 4\). Finally, multiply 4 by \(-3\) and add: \(4 \times -3 = -12\), \, \(112 - 12 = 100\).
04

Interpret the Remainder

The last number (100) is the remainder of the division, which by the Remainder Theorem tells us \(P(-3) = 100\).

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Key Concepts

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

Polynomial Division
Polynomial division, much like regular division, allows us to divide one polynomial by another. It helps simplify problems and solve for roots or evaluate polynomials. When it comes to complex polynomials like those of higher degrees, the process can be a bit overwhelming. But synthetic division simplifies this, especially when dividing by linear divisors of the form \(x - c\).

Here's how synthetic division works:
  • Only applicable when dividing by linear divisors.
  • Utilizes only the coefficients of the polynomial, making calculations simpler.
  • Helps in finding polynomial remainders efficiently.
Synthetic division streamlines polynomial division, reducing tedious calculations and allowing for faster problem-solving.
Remainder Theorem
The Remainder Theorem is a key concept in polynomial mathematics. It simplifies the process of evaluating polynomials without plugging in variables directly. This theorem states that when you divide a polynomial \(P(x)\) by \(x - c\), the remainder of this division is equal to \(P(c)\).

This means:
  • If you need to find \(P(c)\), you can simply use synthetic division to find the remainder.
  • It eliminates the need for long calculations or substitutions.
  • This theorem is especially useful for solving polynomial equations and verifying roots.
It provides a method that is less prone to errors compared to direct substitution, especially for high-degree polynomials.
Evaluating Polynomials
Evaluating polynomials is the process of finding the value of a polynomial for a given variable. This essential skill allows us to understand how polynomials behave and determine their values at specific points. With the Remainder Theorem, evaluation becomes easier and more efficient.

When given a polynomial \(P(x)\) and a specific value \(c\), we aim to find \(P(c)\). By performing synthetic division with \(x - c\), we can directly find the remainder, which is the value \(P(c)\).

Benefits of using synthetic division for evaluating polynomials:
  • It provides quick and easy calculations.
  • Useful for checking if a particular number is a root of the polynomial.
  • Simplifies the process of solving polynomial functions at given points.
This allows us to tackle complex polynomial problems with ease and precision.
Polynomial Coefficients
Polynomial coefficients are the numbers that multiply the powers of the variable in a polynomial. Understanding and utilizing these coefficients is critical in performing operations like synthetic division effectively.

Here's what coefficients tell us:
  • They define the terms of a polynomial such as \(-2\) in \(-2x^6\) or \(40\) in \(40x^4\).
  • In synthetic division, these coefficients are arranged in order, making calculations direct and structured.
  • Any missing terms (like \(x^3\) in our example) should be represented by a zero coefficient to maintain the polynomial's integrity and order.
Correctly identifying and using coefficients supports the simplified mechanics of synthetic division, and helps ensure accurate results during polynomial evaluations.

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Most popular questions from this chapter

Volume of a Rocket A rocket consists of a right circular cylinder of height 20 \(\mathrm{m}\) surmounted by a cone whose height and diameter are equal and whose radius is the same as that of the cylindrical section. What should this radius be (rounded to two decimal places) if the total volume is to be 500\(\pi / 3 \mathrm{m}^{3}\) ?

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=2 x^{3}+5 x^{2}+x-2 ; \quad a=-3, b=1 $$

The Doppler Effect As a train moves toward an observer (see the figure), the pitch of its whistle sounds higher to the ob- server than it would if the train were at rest, because the crests of the sound waves are compressed closer together. This phenomenon is called the Doppler effect. The observed pitch \(P\) is a function of the speed \(v\) of the train and is given by $$ P(v)=P_{0}\left(\frac{s_{0}}{s_{0}-v}\right) $$ where \(P_{0}\) is the actual pitch of the whistle at the source and \(S_{0}=332 \mathrm{m} / \mathrm{s}\) is the speed of sound in air. Suppose that a train has a whistle pitched at \(P_{0}=440 \mathrm{Hz}\) . Graph the func- tion \(y=P(v)\) using a graphing device. How can the vertical asymptote of this function be interpreted physically?

The real solutions of the given equation are rational. List all possible rational roots using the Rational Zeros Theorem, and then graph the polynomial in the given viewing rectangle to determine which values are actually solutions. (All solutions can be seen in the given viewing rectangle.) $$ x^{4}-5 x^{2}+4=0 ;[-4,4] \mathrm{by}[-30,30] $$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(\mathrm{a}) .\) $$ P(x)=4 x^{5}-18 x^{4}-6 x^{3}+91 x^{2}-60 x+9 $$
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