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Chapter 7: Problem 55

In Exercises \(51-62,\) evaluate each polynomial for the given values. $$6 x^{2}-7 x+3 \text { for } x=-\frac{1}{2}$$

Short Answer

Expert verified
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Step by step solution

01

Substitute the given value

Substitute the value \(x = -\frac{1}{2}\) into the polynomial \(6x^2 - 7x + 3\).
02

Compute \(6x^2\)

Calculate \(6 \left(-\frac{1}{2}\right)^2\). First, compute \(\left(-\frac{1}{2}\right)^2 = \frac{1}{4}\). Then, multiply this by 6 to get \(6 \times \frac{1}{4} = \frac{6}{4} = \frac{3}{2}\).
03

Compute \(-7x\)

Calculate \(-7 \left(-\frac{1}{2}\right)\). Multiply to get \(-7 \times -\frac{1}{2} = \frac{7}{2}\).
04

Add the constant term

The constant term is 3. So, you have \(\frac{3}{2} + \frac{7}{2} + 3\).
05

Perform the addition

Combine the fractions and the constant term: \(\frac{3}{2} + \frac{7}{2} = \frac{10}{2} = 5\). Then, add 3 to get \(5 + 3 = 8\).

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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 technique used to replace variables with given values. This allows us to evaluate expressions or polynomials effectively. In our exercise, we are given the polynomial \(6x^2 - 7x + 3\) and need to find its value when \(x = -\frac{1}{2}\). To do this using the substitution method:
  • Identify the given value you need to substitute.
  • Replace every occurrence of the variable in the polynomial with this value.
  • Proceed to simplify and calculate the final result.
    • By replacing \(x\) with \(-\frac{1}{2}\) in the polynomial, we'll perform calculations to eventually reach the solution.
polynomial evaluation
Polynomial evaluation is the process of determining the value of a polynomial function for a given value of its variable(s). In mathematical terms, given a polynomial \(P(x) = 6x^2 - 7x + 3\), we can find \(P(-\frac{1}{2})\) by substituting \(x = -\frac{1}{2}\). Here's how to carry out polynomial evaluation:
  • After substituting the given value, follow arithmetic operations strictly according to the order of operations (PEMDAS/BODMAS rules).
  • Begin with exponents, then proceed to multiplication and division from left to right.
  • Finally, complete addition and subtraction operations.
    • By carefully following these steps, you ensure accurate results in polynomial evaluation.
simplifying fractions
Simplifying fractions plays a key role in evaluating polynomials, especially when dealing with fractional values. During our exercise, you will encounter fractions throughout the simplification process. The steps are clear:
  • Complete multiplication/division as necessary before simplifying.
  • Look for common factors in the numerator and denominator to reduce the fraction.
  • Convert complex fractions into simpler terms when adding or subtracting.
    • For example, when calculating \(6 \times (-\frac{1}{2})^2\), we computed \((-\frac{1}{2})^2 = \frac{1}{4}\), then multiplied it by 6 to get \(\frac{6}{4} = \frac{3}{2}\). Breaking fractions down accurately helps avoid mistakes and streamline calculations.
basic algebra steps
Basic algebra forms the groundwork for solving polynomial evaluations and includes several fundamental steps:
  • Identify and substitute values correctly to start computations.
  • Adhere to order of operations, simplifying exponents and performing arithmetic in sequence.
  • Simplify fractions by finding common denominators and reducing them correctly.
    • Integrating all these steps proves vital in our example. Substituting \(x = -\frac{1}{2}\):
    • First, calculate \(6(-\frac{1}{2})^2 = \frac{3}{2}\).
    • Then, compute \(-7(-\frac{1}{2}) = \frac{7}{2}\).
    • Finally, add up \(\frac{3}{2} + \frac{7}{2} + 3 = 8\).
      • Understanding these basic algebra steps ensures a solid foundation for evaluating more complex expressions.

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Most popular questions from this chapter

Do your computation using scientific notation. $$\frac{(2400)(1500)}{(90,000)(4000)}$$

Estimate the answer without actually carrying out the computation and make the most appropriate choice. If you multiply \(1.45 \times 10^{-6}\) by \(2.03 \times 10^{4},\) the result is closest to (a) 10 (b) 100 (c) 1 (d) 0.1 (e) 0.01

Convert each number into standard notation. $$1.76 \times 10^{5}$$

Solve for x. $$-2<3 x+4 \leq 16$$

In this section we defined \(a^{0}=1 .\) It is important to realize that a definition is neither right nor wrong-it just is. The proper question to ask about a definition is "Is it useful?" What happens if someone decides on an "alternative definition" such as \(a^{0}=7\) because 7 happens to be his or her favorite number? What happens when we consider the expression \(a^{0} \cdot a^{4} ?\) If you use exponent rule 1 you get \(a^{0} \cdot a^{4}=a^{0+4}=a^{4} .\) However, if you use this alternative definition, you get \(a^{0} \cdot a^{4} \stackrel{\underline{2}}{=} 7 a^{4} .\) The answer we get from the alternative definition is not consistent with the answer we get from the exponent rule. The exponent rules and this alternative definition cannot coexist. Since we do not want to throw away all the exponent rules, we must modify the alternative definition so that it is consistent with all the exponent rules. Verify that our definition of \(a^{0}=1\) is consistent with all five exponent rules.
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