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Chapter 5: Problem 69

Evaluate polynomial function for \(x=-1\). \(f(x)=-5 x^{3}+3 x^{2}-4 x-3\)

Short Answer

Expert verified
f(-1) = 9.

Step by step solution

01

Identify the Polynomial

The polynomial function to be evaluated is given by f(x) = -5x^{3} + 3x^{2} - 4x - 3.
02

Substitute the Value of x

To evaluate the polynomial function for x = -1, substitute -1 into the polynomial: f(-1) = -5(-1)^{3} + 3(-1)^{2} - 4(-1) - 3.
03

Compute Each Term Separately

Calculate each term of the substituted polynomial: -5(-1)^{3} = -5(-1) = 5 3(-1)^{2} = 3(1) = 3 -4(-1) = 4 -3 remains as -3.
04

Add the Computed Terms Together

Combine the results from Step 3: 5 + 3 + 4 - 3.
05

Simplify the Expression

Add the values together to find the final result: 5 + 3 + 4 - 3 = 9.

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Key Concepts

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

Polynomial Evaluation
In mathematics, evaluating a polynomial means finding the value of the polynomial for a given value of the variable. For instance, if we have a polynomial function like \(f(x) = -5x^3 + 3x^2 - 4x - 3\), and we want to evaluate it for \(x = -1\), we are essentially calculating \(f(-1)\). Polynomials are just expressions that involve variables raised to whole number powers and their coefficients. They look complex, but with a systematic approach, you can easily evaluate them. This brings us to the substitution method.
Substitution Method
The substitution method is a straightforward way to evaluate polynomials. Here鈥檚 how you can do it step-by-step:

First, take the polynomial and substitute the given value of the variable (in this case \(x = -1\)) into every instance of the variable in the polynomial.

So, if our polynomial is \(f(x) = -5x^3 + 3x^2 - 4x - 3\), we substitute \(-1\) for \(x\):

\(f(-1) = -5(-1)^3 + 3(-1)^2 - 4(-1) - 3\).

This substitution transforms the polynomial into a numerical expression that you can easily calculate.
Simplifying Expressions
Once you have substituted the value into the polynomial, the next step is to simplify the expression. Here's how you can do it:

  • Calculate each term separately:

  • \(-5(-1)^3 = -5(-1) = 5\)

  • \(3(-1)^2 = 3(1) = 3\)

  • \(-4(-1) = 4\)

  • \(-3\ \text{remains as} \ -3\).

Now, add the individual results of these terms together:

\(5 + 3 + 4 - 3\).

The last step is to simplify this numerical expression by performing basic addition and subtraction:

\(5 + 3 + 4 - 3 = 9\).

This final value is the result of evaluating the polynomial for \(x = -1\).

By following these steps, you can easily evaluate any polynomial for any given value of the variable!

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