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Chapter 5: Problem 8

For each polynomial function, find (a) \(f(-1)\) and \((b) f(2)\). \(f(x)=-x^{2}-x^{3}+11 x\)

Short Answer

Expert verified
f(-1) = -11, f(2) = 10

Step by step solution

01

- Substitute x = -1 into the polynomial

To find f(-1) , replace every x in the polynomial f(x) = -x^2 - x^3 + 11x with -1 . f(-1) = -(-1)^2 - (-1)^3 + 11(-1)
02

- Simplify the expression for f(-1)

Calculate each term: (-1)^2 = 1, -(-1)^2 = -1, (-1)^3 = -1, -(-1)^3 = 1, 11(-1) = -11. So, f(-1) = -1 + 1 - 11 = -11.
03

- Substitute x = 2 into the polynomial

To find f(2) , replace every x in the polynomial f(x) = -x^2 - x^3 + 11x with 2. f(2) = -(2)^2 - (2)^3 + 11(2).
04

- Simplify the expression for f(2)

Calculate each term: (2)^2 = 4, -(2)^2 = -4, (2)^3 = 8, -(2)^3 = -8, 11(2) = 22. So, f(2) = -4 - 8 + 22 = 10.

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Key Concepts

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

Substitution Method
The substitution method is a key technique for evaluating polynomial functions. To use this method, you replace the variable, usually denoted as x, with a specific number provided in the problem. This allows you to simplify the expression and find the value of the polynomial for that particular input.
For the given problem, we need to find the values of the polynomial function when x is -1 and then when x is 2. By substituting these values in place of x, we will be able to calculate the specific outputs of the function.
Polynomial Functions
Polynomial functions are expressions consisting of variables and coefficients, encompassing terms in the form of ax^n, where a is a constant, x is a variable, and n is a non-negative integer. The given polynomial function in this exercise is:
f(x) = -x^2 - x^3 + 11x
This polynomial is a combination of a quadratic term (-x^2), a cubic term (-x^3), and a linear term (11x). Each term must be evaluated separately before the results are combined to find the final value of the function for any given x.
Function Evaluation
Function evaluation involves substitifying a value into the function and performing the necessary arithmetic operations to obtain a result. Let鈥檚 look at how function evaluation works through our steps:
1. **Substitute x = -1**:
Let's replace x with -1:
f(-1) = -(-1)^2 - (-1)^3 + 11(-1)
2. **Simplify**:
(-1)^2 = 1, -(-1)^2 = -1, (-1)^3 = -1, -(-1)^3 = 1, 11(-1) = -11.
So, f(-1) = -1 + 1 - 11 = -11.

3. **Substitute x = 2**:
Now replace x with 2:
f(2) = -(2)^2 - (2)^3 + 11(2)
4. **Simplify**:
(2)^2 = 4, -(2)^2 = -4, (2)^3 = 8, -(2)^3 = -8, 11(2) = 22.
So, f(2) = -4 - 8 + 22 = 10.
Negative Exponents
Negative exponents often pose a challenge, but it's quite simple once you understand the rule. A negative exponent indicates that you should take the reciprocal of the base and then apply the positive exponent. However, in the given polynomial function, there are no negative exponents, we only dealt with negative coefficients.
Let鈥檚 recap with an example: if you had an expression like x^(-2), you would rewrite it as 1/x^2.
While simplifying, ensure you correctly apply the rules for your coefficients and exponents to avoid errors.

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Most popular questions from this chapter

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