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Chapter 5: Problem 39

Find the value of each polynomial for }(a) x=2 \text { and }(b) x=-1 $$ -3 x^{2}+14 x-2 $$

Short Answer

Expert verified
For x=2, the polynomial evaluates to 14. For x=-1, it evaluates to -19.

Step by step solution

01

Substitute x=2 into the polynomial

First, take the given polynomial -3x^{2} + 14x - 2 . Substitute x = 2 : -3(2)^{2} + 14(2) - 2 .
02

Calculate the value for x=2

Evaluate the expression: -3(4) + 14(2) - 2 = -12 + 28 - 2 = 14 .
03

Substitute x=-1 into the polynomial

Next, reexamine the polynomial -3x^{2} + 14x - 2 . Substitute x = -1 : -3(-1)^{2} + 14(-1) - 2 .
04

Calculate the value for x=-1

Evaluate the expression: -3(1) + 14(-1) - 2 = -3 - 14 - 2 = -19 .

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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 involves replacing a variable in an algebraic expression with a specific value. This is a common technique used to evaluate expressions and solve equations. In this exercise, we substitute the values of \(x\) into the polynomial to find its value.
  • Identify the variable in the polynomial or equation.
  • Replace the variable with the given value.
  • Simplify the resulting expression to find the answer.
For example, given the polynomial \(-3x^{2} + 14x - 2\), we replace \(x\) with 2 and simplify:
\[ -3(2)^{2} + 14(2) - 2 \]
This method helps in determining the exact value of the polynomial for specific variable inputs.
Polynomial Evaluation
Polynomial evaluation is the process of computing the value of a polynomial for a given value of the variable. This involves substituting and simplifying as demonstrated in the steps.
Here are the general steps:
  • Substitute the value of the variable into the polynomial.
  • Follow the order of operations (parentheses, exponents, multiplication, division, addition, and subtraction).
  • Calculate the final value.
For example, substituting \(x = -1\) in the polynomial \(-3x^{2} + 14x - 2\):
\[ -3(-1)^{2} + 14(-1) - 2 \]
Simplify it step-by-step to get the final value. This systematic process helps in clearly evaluating polynomials.
Algebraic Expressions
Algebraic expressions consist of numbers, variables, and arithmetic operations. Polynomials are a type of algebraic expression that includes terms with variables raised to whole number exponents.
Understanding different parts of an algebraic expression is pivotal:
  • Terms: Separate elements in an expression, e.g., \(-3x^{2}\), \(14x\), and \(-2\) in \(-3x^{2} + 14x - 2\).
  • Coefficients: Numbers multiplied by variables, such as -3 and 14 in the polynomial.
  • Constants: Numbers on their own, such as -2.
  • Exponents: Powers to which variables are raised, such as the 2 in \(x^{2}\).
Recognizing and understanding the components of algebraic expressions can make the substitution method and polynomial evaluation more straightforward.

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Most popular questions from this chapter

Perform each division. $$ \left(2 x^{3}+x+2\right) \div(x+1) $$

List all positive integer factors of each number. $$ 23 $$

Perform each division. $$ \left(3 x^{3}+x+5\right) \div(x+1) $$

Use scientific notation to calculate the answer to each problem. Astronomers using the Spitzer Space Telescope discovered a twisted double- helix nebula, a conglomeration of dust and gas stretching across the center of the Milky Way galaxy. This nebula is \(25,000\) light-years from Earth. If one light-year is about \(6,000,000,000,000\) (that is, 6 trillion) miles, about how many miles is the twisted double-helix nebula from Earth?

List all positive integer factors of each number. $$ 18 $$
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