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Chapter 3: Problem 44

Determine whether each given number is a zero of the polynomial function following the mumber. $$3, G(r)=r^{4}+4 r^{3}+5 r^{2}+3 r+17$$

Short Answer

Expert verified
3 is not a zero of the polynomial function.

Step by step solution

01

- Substitute the Number into the Polynomial

Substitute the given number, which is 3, into the polynomial function \( G(r) = r^{4} + 4r^{3} + 5r^{2} + 3r + 17 \). So this will be \( G(3) = 3^{4} + 4(3)^{3} + 5(3)^{2} + 3(3) + 17 \).
02

- Calculate Each Term

Calculate the value of each term separately: \( 3^{4} = 81 \), \( 4 \times 3^{3} = 4 \times 27 = 108 \), \( 5 \times 3^{2} = 5 \times 9 = 45 \), \( 3 \times 3 = 9 \), and \( 17 = 17 \).
03

- Add the Results

Now add up all the results obtained: \( 81 + 108 + 45 + 9 + 17 \).
04

- Determine the Sum

Perform the addition: \( 81 + 108 = 189 \), \( 189 + 45 = 234 \), \( 234 + 9 = 243 \), \( 243 + 17 = 260 \).
05

- Conclusion

Since the result is not zero, \( G(3) = 260 \), this means that 3 is not a zero of the polynomial function.

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Key Concepts

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

Polynomial Evaluation
Polynomial evaluation focuses on calculating the value of a polynomial at a specific point. Let's break down how you do it:

First, you have a polynomial function like: \$G(r) = r^{4} + 4r^{3} + 5r^{2} + 3r + 17\. A polynomial is just a mathematical expression that involves a sum of powers of a variable.

To evaluate this polynomial at a particular value of \(r\), simply substitute that value for \(r\) in the function. In our exercise, we substitute the value 3, resulting in: \$G(3) = 3^{4} + 4(3)^{3} + 5(3)^{2} + 3(3) + 17\.

This involves simple arithmetic operations:
  • Calculate each exponent part individually.
  • Multiply the coefficients by the results of those exponentiations.

Finally, add all these individual results together to get the value of the polynomial at \(r=3\).
Finding Polynomial Zeros
Finding zeros of a polynomial means figuring out where the polynomial equals zero. Here's how to think about it:

Let's say you have the polynomial function, \$G(r) = r^{4} + 4r^{3} + 5r^{2} + 3r + 17\.

To find its zeros, we need to determine the values of \(r\) that make \$G(r) = 0\.

In simple terms, a zero is any number that you can substitute into the polynomial to get zero.

So, if you substitute \(r = 3\) into the polynomial and calculate, you should get zero for it to be a zero:
  • Substitute \(r = 3\): \$G(3) = 3^{4} + 4(3)^{3} + 5(3)^{2} + 3(3) + 17\
  • Calculate the value: \(81 + 108 + 45 + 9 + 17 = 260\)

Since 260 is not zero, \(r = 3\) is not a zero of the polynomial.
Substituting Values in Polynomials
Substituting values in polynomials is a straightforward process. Here's how you do it:
  • Take the given polynomial: \$G(r) = r^{4} + 4r^{3} + 5r^{2} + 3r + 17\
  • Identify the value you'll substitute into the polynomial (in this case, 3).

Now, follow these steps:

  • Replace every instance of \(r\) in the polynomial with 3.
  • Perform all the exponent calculations first: \(3^{4} = 81\), \(3^{3} = 27\), \(3^{2} = 9\).
  • Multiply these results by their respective coefficients: \(4 \times 27 = 108\), \(5 \times 9 = 45\), and so on.

Finally, add up all these results together to get the final value:
  • \(81 + 108 + 45 + 9 + 17 = 260\).

This shows how substitution works, and helps you find if a number is a zero or not.

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Most popular questions from this chapter

Use Descartes's rule of signs to discuss the possibilities for the roots of the equation \(-x^{4}-6 x^{2}-3 x+9=0\)

Find the additive inverse and multiplicative inverse of \(-8 i\).

An engineer is designing a cylindrical metal tank that is to hold \(500 \mathrm{ft}^{3}\) of gasoline. a. Write the height \(h\) as a function of the radius \(r\) Hint: The volume is given by \(V=\pi r^{2} h\) b. Use the result of part (a) to write the surface area \(S\) as a function of \(r\) and graph it. Hint: The surface area is given by \(S=2 \pi r^{2}+2 \pi r h\) c. Use the minimum feature of a graphing calculator to find the radius to the nearest tenth of a foot that minimizes the surface area. Ignore the thickness of the metal. d. If the tank costs 8 dollar per square foot to construct, then what is the minimum cost for making the tank?

Solve each polynomial inequality using the test-point method. $$x^{4}-19 x^{2}+90 \leq 0$$

Solve each polynomial inequality using the test-point method. $$x^{3}-3 x>0$$
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