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Chapter 5: Problem 59

Evaluate polynomial for \(x=3\) and for \(x=-3\). \(-2 x^{3}-3 x^{2}+4 x+2\)

Short Answer

Expert verified
When x = 3, the polynomial evaluates to -67. When x = -3, it evaluates to 17.

Step by step solution

01

Identify the Polynomial

The given polynomial to evaluate is a(x) = -2x^3 - 3x^2 + 4x + 2
02

Substitute x = 3 into the Polynomial

Replace x with 3 in the polynomial: a(3) = -2(3)^3 - 3(3)^2 + 4(3) + 2
03

Calculate Each Term for x = 3

Evaluate each term individually: -2(3)^3 = -2(27) = -54, -3(3)^2 = -3(9) = -27, 4(3) = 12, and simply 2 Thus: a(3) = -54 - 27 + 12 + 2
04

Add the Results for x = 3

Sum the evaluated terms: -54 - 27 + 12 + 2 = -67
05

Substitute x = -3 into the Polynomial

Replace x with -3 in the polynomial: a(-3) = -2(-3)^3 - 3(-3)^2 + 4(-3) + 2
06

Calculate Each Term for x = -3

Evaluate each term individually: -2(-3)^3 = -2(-27) = 54, -3(-3)^2 = -3(9) = -27, 4(-3) = -12, and simply 2 Thus: a(-3) = 54 - 27 - 12 + 2
07

Add the Results for x = -3

Sum the evaluated terms: 54 - 27 - 12 + 2 = 17

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Key Concepts

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

Polynomial Evaluation
Polynomial evaluation is the process of calculating the value of a polynomial function at a particular point. Polynomials are expressions consisting of variables and coefficients, combined using addition, subtraction, and multiplication. In this exercise, we evaluate the polynomiala(x) = -2x^3 - 3x^2 + 4x + 2at two different points, x = 3 and x = -3. This helps us see how the polynomial behaves at these specific values.
To evaluate the given polynomial, we need to substitute the value of x into the polynomial and simplify the expression step-by-step, as shown in the provided solution.
Let's break down the process in the upcoming sections to ensure a clearer understanding.
Substitution Method
The substitution method in polynomial evaluation involves replacing the variable x with a given numerical value. This allows us to calculate the polynomial's value at that specific point.
The steps are straightforward:
  • Identify the polynomial expression, which in our case is a(x) = -2x^3 - 3x^2 + 4x + 2.
  • Replace x with the given value.
  • Calculate each term individually by following the order of operations (PEMDAS/BODMAS).
Let's see this in action for x = 3:
  • Replace x with 3: a(3) = -2(3)^3 - 3(3)^2 + 4(3) + 2.
  • Calculate each term: -2(3)^3 = -54, -3(3)^2 = -27, 4(3) = 12, and 2 remains as 2.
  • Add the results: -54 - 27 + 12 + 2 = -67.
This method makes the computation organized and easier to follow.
Algebraic Calculation
Algebraic calculation in polynomial evaluation involves systematically simplifying the polynomial after substitution. Each term is calculated by following mathematical rules of operations.
Let's evaluate the same polynomial for x = -3:
  • First, substitute -3 into the polynomial: a(-3) = -2(-3)^3 - 3(-3)^2 + 4(-3) + 2.
  • Next, calculate each term: -2(-3)^3 = 54, -3(-3)^2 = -27, 4(-3) = -12, and 2 remains 2.
  • Finally, sum the evaluated terms: 54 - 27 - 12 + 2 = 17.
By performing these algebraic operations accurately, we find the value of the polynomial at the given points. The detailed steps help avoid mistakes and ensure the correct result. Remember, practice makes perfect, so try evaluating more polynomials using these methods!

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