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Chapter 3: Problem 41

Determine whether each given number is a zero of the polynomial function following the mumber. $$-2, g(x)=3 x^{3}-6 x^{2}-3 x-19$$

Short Answer

Expert verified
-2 is not a zero of the polynomial function.

Step by step solution

01

- Substitute the Number

Substitute \(x = -2\) into the polynomial function \(g(x) = 3x^3 - 6x^2 - 3x - 19\).
02

- Calculate Each Term

Calculate the value of each term for \(x = -2\): \[3(-2)^3 - 6(-2)^2 - 3(-2) - 19\].
03

- Simplify the Expression

Simplify the expression:\[3(-8) - 6(4) + 6 - 19\].
04

- Evaluate the Sum

Evaluate the sum:\[-24 - 24 + 6 - 19\] and simplify further to\[-24 - 24 + 6 - 19 = -61\].
05

- Conclusion

Since \(g(-2) = -61\), which is not 0, \(-2\) is not a zero of the polynomial function \(g(x) = 3x^3 - 6x^2 - 3x - 19\).

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Key Concepts

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

polynomial functions
A polynomial function is an algebraic expression made up of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. In our example, the given polynomial function is:
\[ g(x) = 3x^3 - 6x^2 - 3x - 19 \]
Here, the highest power of the variable (x) is 3, making it a cubic polynomial. Polynomials are a fundamental concept in algebra because they describe a wide range of functions and are used in many real-world applications, including physics, economics, and engineering.
evaluating functions
Evaluating a function involves finding the value of the function at a particular value of its variable. For instance, to determine whether \( -2 \) is a zero of \( g(x) = 3x^3 - 6x^2 - 3x - 19 \), we need to evaluate \( g(-2) \). This means substituting \(-2 \)for \( x \) in the equation.
  • Function: \[ g(x) = 3x^3 - 6x^2 - 3x - 19 \]
  • Evaluate at \(x = -2 \): \[ g(-2) \]
In mathematical terms, evaluating a polynomial function follows a specific order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
substitution method
The substitution method involves replacing the variable in the polynomial function with the given number. For our problem, this entails substituting \(-2 \)into \( g(x) \). Here are the steps:
  • Start with the function: \[ g(x) = 3x^3 - 6x^2 - 3x - 19 \]
  • Substitute \( x = -2 \): \[ 3(-2)^3 - 6(-2)^2 - 3(-2) - 19 \]
After substitution, we calculate each term one by one: \[ 3(-2)^3 - 6(-2)^2 - 3(-2) - 19 \]. This method helps in identifying whether the given number is a zero of the polynomial. If the result after substitution equals zero, the number is a zero of the polynomial function.
simplification
After substituting the value, the next step is to simplify the expression. Simplification makes complex expressions easier to understand by performing arithmetic operations step-by-step:
  • Calculate each term: \[ 3(-2)^3 = 3(-8) = -24 \]
  • \[ -6(-2)^2 = -6(4) = -24 \]
  • \[ -3(-2) = 6 \]
  • Combine everything: \[ -24 - 24 + 6 - 19 = -61 \]
After calculating each term, we sum them up to get the simplified value \( -61 \). Since this value is not zero, \( -2 \) is not a root of \( g(x) \). Simplification is crucial in verifying whether the given number is indeed a zero of the polynomial or not.

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