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Chapter 5: Problem 5

For each polynomial function, find (a) \(f(-1)\) and \((b) f(2)\). \(f(x)=5 x^{4}-3 x^{2}+6\)

Short Answer

Expert verified
f(-1) = 8, f(2) = 74

Step by step solution

01

- Understand the Problem

We need to evaluate the polynomial function at two different points: \(-1\) and \(2\). The polynomial function provided is \(f(x) = 5x^4 - 3x^2 + 6\).
02

- Substitute \(x = -1\) and Simplify

Replace \(x\) with \( -1\) in the polynomial function and simplify: \[ f(-1) = 5(-1)^4 - 3(-1)^2 + 6 \] Calculate each term: \[ (-1)^4 = 1 \] \[ (-1)^2 = 1 \] Substitute these values back: \[ f(-1) = 5(1) - 3(1) + 6 \] Simplify the expression: \[ f(-1) = 5 - 3 + 6 = 8 \]
03

- Substitute \(x = 2\) and Simplify

Replace \(x\) with \( 2\) in the polynomial function and simplify: \[ f(2) = 5(2)^4 - 3(2)^2 + 6 \] Calculate each term: \[ (2)^4 = 16 \] \[ (2)^2 = 4 \] Substitute these values back: \[ f(2) = 5(16) - 3(4) + 6 \] Simplify the expression: \[ f(2) = 80 - 12 + 6 = 74 \]

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Key Concepts

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

polynomial evaluation
When we talk about polynomial evaluation, it means finding the value of a polynomial function for a specific value of the variable. For instance, in the provided exercise, we are asked to find \(f(-1)\) and \(f(2)\) for the polynomial function \(f(x) = 5x^4 - 3x^2 + 6\).
This involves plugging in the values \(x = -1\) and \(x = 2\) into the polynomial and then simplifying the expressions.
By evaluating polynomials at given points, we can understand how the polynomial behaves for those inputs. This is a fundamental skill in algebra.
substitution method
The substitution method is straightforward and essential for evaluating polynomial functions and solving equations. Here's how we use it in this exercise:
  • First, identify the polynomial function and the specific values you need to substitute. In our case: \(f(x)=5x^{4}-3x^{2}+6 \).
  • Next, substitute \(x = -1\) into the polynomial. Replace every \(x\) with \(-1\) and simplify: \[ f(-1) = 5(-1)^4 - 3(-1)^2 + 6 \].
  • Follow the same steps for \(x = 2\): \[ f(2) = 5(2)^4 - 3(2)^2 + 6 \].
This method allows us to find the function's value at any given point by substituting the desired number in place of the variable and simplifying the result.
simplification of polynomial expressions
Simplification of polynomial expressions is crucial as it helps us break down complicated expressions into simpler forms for easy evaluation. Let鈥檚 simplify the polynomial expressions from the given exercise:
  • First, for \(f(-1)\): Start by substituting \(-1\) into \(f(x) = 5x^4 - 3x^2 + 6\) and calculate each term:
  • \((-1)^4 = 1\) and \((-1)^2 = 1\)
  • Substitute these values back into the polynomial: \[ f(-1) = 5(1) - 3(1) + 6 = 8 \]. Typically, perform arithmetic operations step-by-step to ensure accuracy:
  • \5 \times 1 = 5, \ [-3 \times 1 = -3,] and \ 5 - 3 + 6 = 8 \.
  • Next, for \ f(2)\: Substitute \ x = 2\, then calculate: \[ (2)^4 = 16, (2)^2 = 4 \]
  • Substituting these values back: \[ f(2) = 5(16) - 3(4) + 6 = 74\] Perform step-by-step simplification:
  • \ 5 \times 16 = 80, -3 \times 4 = -12 \
  • Finally, \ 80 - 12 + 6 = 74 \.
This methodical approach to simplification results in accurate answers and a better grasp of polynomial expressions.

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Most popular questions from this chapter

Two expressions are given. Replace x with 3 and y with 4 to show that, in general, the two expressions are not equivalent. $$ (x+y)^{3} ; \quad x^{3}+y^{3} $$

Write each number in standard notation. $$ 2.54 \times 10^{-3} $$

Evaluate. Express answers in standard notation. $$ \frac{12 \times 10^{4}}{2 \times 10^{6}} $$

Simplify each expression so that no negative exponents appear in the final result. Assume that all variables represent nonzero real numbers. $$ 3 a^{2}\left(-5 a^{-6}\right)(-2 a)^{0} $$

Write each number in standard notation. $$ 5.42 \times 10^{-4} $$
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