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Chapter 5: Problem 60

Evaluate polynomial for \(x=3\) and for \(x=-3\). \(-3 x^{3}+7 x^{2}-4 x-8\)

Short Answer

Expert verified
For x=3, the value is -38. For x=-3, the value is 148.

Step by step solution

01

- Substitute x=3 into the polynomial

Start by substituting the value of x=3 into the polynomial \(-3 x^{3}+7 x^{2}-4 x-8\). This results in \(-3(3)^3 + 7(3)^2 - 4(3) - 8\).
02

- Calculate each term for x=3

Calculate each term separately:\(-3(3)^3 = -3(27) = -81\), \(7(3)^2 = 7(9) = 63\), \(-4(3) = -12\),\(-8 = -8\).
03

- Add the terms for x=3

Sum up all the terms for x=3: \(-81 + 63 - 12 - 8 = -38\).
04

- Substitute x=-3 into the polynomial

Now substitute the value of x=-3 into the polynomial \(-3 x^{3}+7 x^{2}-4 x-8\). This results in \(-3(-3)^3 + 7(-3)^2 - 4(-3) - 8\).
05

- Calculate each term for x=-3

Calculate each term separately:\(-3(-3)^3 = -3(-27) = 81\), \(7(-3)^2 = 7(9) = 63\), \(-4(-3) = 12\),\(-8 = -8\).
06

- Add the terms for x=-3

Sum up all the terms for x=-3: \(81 + 63 + 12 - 8 = 148\).

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Key Concepts

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

substitute values in polynomials
To evaluate a polynomial, we need to substitute the variable with a specific value. This means replacing every instance of the variable in the polynomial with the given number.
For instance, given the polynomial \(-3 x^{3}+7 x^{2}-4 x-8\) and the task to evaluate it for \(x=3\):
  • Replace every x with 3.
  • Substitute values directly into the polynomial equation: \(-3(3)^3 + 7(3)^2 - 4(3) - 8\).
This substitution helps convert the algebraic expression into a simple numerical calculation, making further simplification straightforward.
polynomial terms
Polynomials are composed of different terms, each term being a product of a number (coefficient) and a variable raised to a power. For example, in the polynomial \(-3x^3 + 7x^2 - 4x - 8\):
  • \(-3x^3\) is the term with x cubed and a coefficient of -3
  • \(7x^2\) is the term with x squared and a coefficient of 7
  • \(-4x\) is the linear term with a coefficient of -4
  • \(-8\) is the constant term
Understanding these terms is the key to manipulating and simplifying the polynomial during evaluation.
simplify expressions
Once we've substituted the value into the polynomial, the next step is to simplify the expression. This involves:
  • Calculating the power of each term first. For instance, \(3^3 = 27\) and \((-3)^3 = -27\).
  • Multiplying each result by its coefficient, such as \(-3(27) = -81\).
  • Adding or subtracting the results of each term to get the final value.
For our example, after substituting \(x=3\), we get \(-81 + 63 - 12 - 8 = -38\).
algebraic operations
Algebra involves several operations that help in simplifying polynomials:
  • Addition & Subtraction: In our polynomial, after finding the value of each term, we add or subtract them to get the final result. For x=3, we perform -81 + 63 - 12 - 8.
  • Multiplication: This is required when calculating the coefficients. For 7(3^2), multiply 7 by 9.
  • Exponentiation: This operation is crucial for finding the values of terms like 3^3 and (-3)^3.
Mastery of these basic algebraic operations is essential for polynomial evaluation and other complex algebra topics.

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Most popular questions from this chapter

Simplify. \(\left(3 x^{2}\right)^{3}+4 x^{2} \cdot 4 x^{4}-x^{4}(2 x)^{2}+\left((2 x)^{2}\right)^{3}-\) \(100 x^{2}\left(x^{2}\right)^{2}\)

Simplify. $$ \frac{4.2 \times 10^{8}\left[\left(2.5 \times 10^{-5}\right) \div\left(5.0 \times 10^{-9}\right)\right]}{3.0 \times 10^{-12}} $$

Replace \(\square\) with \(>,<,\) or \(=\) to write a true sentence. $$ 4^{2} \square 4^{3} $$

For each pair of functions \(f\) and \(g\), determine the domain of \(f / g\) $$ \begin{aligned} &f(x)=3 x-2\\\ &g(x)=2 x-8 \end{aligned} $$

Simplify. $$ y^{4 x} \cdot y^{2 x} $$
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