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Chapter 5: Problem 105

Use synthetic division to find the function values. \(g(x)=x^{3}-6 x^{2}+3 x+10 ;\) find \(g(-5)\)

Short Answer

Expert verified
g(-5) = -280

Step by step solution

01

- Set up synthetic division

Write down the coefficients of the polynomial. For the function \(g(x) = x^3 - 6x^2 + 3x + 10\), the coefficients are 1, -6, 3, and 10. Notice that we do not skip any powers of x. Now, set up the synthetic division with \(-5\) on the left and these coefficients on the right:
02

- Bring down the leading coefficient

The first coefficient (1) is brought down. This is the initial value present in our solution.
03

- Multiply and add

Multiply \(-5\) by the value you just brought down (1). This gives \(-5\). Write this result under the next coefficient (-6). Add these two values: \(-6 + (-5) = -11\).
04

- Repeat the multiply and add process

Continue this process for the remaining coefficients:- Multiply \(-5\) by \(-11\): \(-5 \times -11 = 55\). Write this under the next coefficient (3). Add these values: \(3 + 55 = 58\).- Multiply \(-5\) by 58: \(-5 \times 58 = -290\). Write this under the next coefficient (10). Add these values: \(10 + (-290) = -280\).
05

- Final result

The last number in the bottom row is the remainder, which is also the value of \(g(-5)\). Based on the calculations, \(g(-5) = -280\).

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Key Concepts

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

Polynomial Functions
Polynomial functions are expressions involving a sum of powers in one or more variables multiplied by coefficients. An example of a polynomial function is \(g(x) = x^3 - 6x^2 + 3x + 10\). Each term in a polynomial function has a coefficient (like 1, -6, 3, and 10 in our example) and a power of the variable (x).
Polynomials are quite common in mathematics and have various applications.
  • The highest power of the variable in a polynomial defines its degree. In our example, the degree is 3 because the highest power of x is 3.
  • Polynomial functions can be used to model real-world situations and solve practical problems.
Understanding how polynomial functions work and how to manipulate them is crucial for higher-level math.
Coefficients
Coefficients are the numerical components of the terms in a polynomial function. In the polynomial function \(g(x) = x^3 - 6x^2 + 3x + 10\), the coefficients are 1, -6, 3, and 10.
  • The coefficient in front of the highest power of x is called the leading coefficient. In our example, the leading coefficient is 1. Bringing this coefficient down is the first step in synthetic division.
  • Coefficients can be positive or negative and can greatly affect the shape and behavior of the polynomial's graph.
  • These values are essential when performing operations such as synthetic division since they provide the necessary numerical values for calculations.
Having a strong grasp of coefficients helps perform polynomial operations and understand their properties.
Function Evaluation
Function evaluation involves finding the value of a function for a specific input. In our case, we needed to find \(g(-5)\) for the polynomial function \(g(x) = x^3 - 6x^2 + 3x + 10\).
  • Function evaluation can be done by direct substitution or methods like synthetic division. For complex polynomials, synthetic division is a slick way to simplify the process.
  • In synthetic division, we use the specific input (in our case, -5), and apply it to the simplified form of the polynomial to obtain the result.
  • This method is efficient and reduces the risk of errors as computations are straightforward and sequential.
By properly using synthetic division, one can quickly and efficiently evaluate polynomial functions at given points.

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Most popular questions from this chapter

Earthquakes in Oklahoma. The number of earthquakes in Oklahoma has increased dramatically since \(2008 .\) Researchers are noting that as the number of wastewater wells that have resulted from oil and gas operations increases, the number of earthquakes also increases (Sources: U.S. Geological Survey; Oklahoma Geological Survey). The number of earthquakes with a 3.0 or greater magnitude in Oklahoma can be modeled by the exponential function $$ E(x)=20.279(1.785)^{x} $$ where \(x\) is the number of years after \(2009 .\) Find the number of earthquakes with a magnitude greater than 3.0 in \(2010,\) in \(2013,\) and in 2016

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Use a graphing calculator to find the approximate solutions of the equation. $$5 e^{5 x}+10=3 x+40$$

Using only a graphing calculator, determine whether the functions are inverses of each other. $$f(x)=\frac{2 x-5}{4 x+7}, g(x)=\frac{7 x-4}{5 x+2}$$

Solve using any method. $$x\left(\ln \frac{1}{6}\right)=\ln 6$$
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