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Chapter 5: Problem 52

Find \(g(3)\) for each polynomial function. $$ g(x)=2-x+4 x^{2} $$

Short Answer

Expert verified
g(3) = 35

Step by step solution

01

Substitute the value of x

To find the value of \( g(3) \), substitute \( x = 3 \) into the polynomial function \( g(x) = 2 - x + 4x^2 \).
02

Substitute and simplify

Substitute 3 into the function: \( g(3) = 2 - 3 + 4(3)^2 \). Simplify the expression step-by-step. First, calculate the exponent: \( 4(3)^2 = 4(9) = 36 \).
03

Combine the terms

Now combine all the terms: \( g(3) = 2 - 3 + 36 \).
04

Final simplification

Simplify the expression fully: \( g(3) = 2 - 3 + 36 = 35 \).

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Key Concepts

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

Understanding Polynomial Expressions
A polynomial expression is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it is made up of terms like constant numbers, variables, and variables raised to a power. For example, the polynomial function given in the exercise is: \( g(x) = 2 - x + 4x^2 \). In this function:
  • 2 is a constant term
  • -x is a linear term
  • 4x^2 is a quadratic term
Each term represents a different part of the polynomial, and understanding these terms helps us evaluate the function for any given number.
Using the Substitution Method
The substitution method is a key technique used to evaluate polynomial expressions. It involves replacing the variable in the expression with a given value. Here's how it works in our example: To find \( g(3) \), substitute \( x = 3 \) into the polynomial function \( g(x) = 2 - x + 4x^2 \). This gives us:
\[ g(3) = 2 - 3 + 4(3)^2 \] By substituting x with 3, we turn the polynomial into an arithmetic expression that can be simplified step-by-step.
Simplification Steps
Simplifying an expression involves performing arithmetic operations step-by-step to reach the final result. After substituting the value, we need to calculate any exponents first, followed by multiplication, and then handle addition or subtraction. For our example:
  • First, calculate the exponent: \( (3)^2 = 9 \)
  • Next, multiply by 4: \( 4 \times 9 = 36 \)
  • Then simplify the expression: \( g(3) = 2 - 3 + 36 \)
  • Finally, combine the terms: 2 - 3 is -1, and -1 + 36 is 35
This gives us the final value: \( g(3) = 35 \)
Performing Function Evaluation
Function evaluation is the process of finding the output of a function for a specific input. The key is to follow the steps systematically:
  • Understand the function given
  • Identify the value to substitute into the function
  • Replace the variable with the given value (substitution method)
  • Simplify the resulting expression step-by-step
  • Arrive at the final answer
In this exercise, we evaluated the function \( g(x) = 2 - x + 4x^2 \) at \( x = 3 \), and found that \( g(3) = 35 \). This approach can be applied to any polynomial function to determine its value for any specific input.

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