Problem 38
Question
In Exercises 25 - 30, prove the inequality for the indicated integer values of \( n \). A factor of \( \left(n^3 - n + 3\right) \) is \( 3 \).
Step-by-Step Solution
Verified Answer
The statement is incorrect, 3 is not a factor of the expression \(n^3 - n + 3\) for \(n = 3\).
1Step 1: Evaluate the expression at n = 3
Substitute \(n = 3\) into the expression \(n^3 - n + 3\). If the expression equals zero, then 3 is a factor as per the factor theorem. Calculate \(3^3 - 3 + 3 = 27 - 3 + 3 = 27 \neq 0\)
2Step 2: Check divisibility by 3
Since the result in Step 1 was not zero, we move on to check if the expression is divisible by 3. But we already obtained a result of 27 which is not divisible by 3, so 3 is not a factor of the expression \(n^3 - n + 3\) for \(n = 3\).
Key Concepts
Polynomial DivisibilityInteger FactorizationProof of Inequality
Polynomial Divisibility
Understanding how to determine if one polynomial is divisible by another is key to exploring algebraic expressions and theorems such as the Factor Theorem. Polynomial divisibility refers to the ability of a polynomial to be divided by another polynomial without leaving a remainder. When considering polynomial divisibility, it's similar to checking if an integer is divisible by another integer.
For example, given a polynomial like \( n^3 - n + 3 \), you might want to know if there is a polynomial factor that divides this evenly, just as you might want to know if an integer divides another integer without a remainder. The Factor Theorem states that if a polynomial \( f(n) \) is divisible by \( (n - a) \), where 'a' is a constant, then \( f(a) \) must equal zero. This means that 'a' is a root of the polynomial.
In the exercise presented, checking if 3 is a factor of \( n^3 - n + 3 \) by evaluating the expression at \( n=3 \) did not yield zero, implying that 3 is not a factor of the polynomial for \( n = 3 \). The common misunderstanding may occur when students think that because \( 3 \) is present in the expression itself, it would automatically be a factor, but polynomial divisibility requires the value of the expression to equal zero when \( n \) is substituted by the factor being tested.
For example, given a polynomial like \( n^3 - n + 3 \), you might want to know if there is a polynomial factor that divides this evenly, just as you might want to know if an integer divides another integer without a remainder. The Factor Theorem states that if a polynomial \( f(n) \) is divisible by \( (n - a) \), where 'a' is a constant, then \( f(a) \) must equal zero. This means that 'a' is a root of the polynomial.
In the exercise presented, checking if 3 is a factor of \( n^3 - n + 3 \) by evaluating the expression at \( n=3 \) did not yield zero, implying that 3 is not a factor of the polynomial for \( n = 3 \). The common misunderstanding may occur when students think that because \( 3 \) is present in the expression itself, it would automatically be a factor, but polynomial divisibility requires the value of the expression to equal zero when \( n \) is substituted by the factor being tested.
Integer Factorization
Integer factorization is the decomposition of a composite number into a product of smaller integers. These smaller integers are factors or divisors of the number. The process is akin to breaking down a number into its building blocks; for example, the number 12 can be factorized into 2 × 2 × 3, where 2 and 3 are prime factors.
When factorizing integers, there are several methods, including trial division, using the sieve of Eratosthenes for smaller numbers, or more complex algorithms for larger numbers. Factorization plays a crucial role when dealing with polynomials as well, where we often seek factors of integer coefficients within the polynomial. However, the Factor Theorem aids in finding factors that are not immediately obvious.
In the original exercise, checking divisibility of \( n^3 - n + 3 \) by the integer 3 can be addressed with integer factorization methods. Yet, it is essential to recognize that in the context of polynomials, we are not simply looking for integer factors but rather polynomial factors. Seeing that the exercise asks to prove inequality, rather than just factorization, suggests students may need to consider how factorization could apply to inequalities.
When factorizing integers, there are several methods, including trial division, using the sieve of Eratosthenes for smaller numbers, or more complex algorithms for larger numbers. Factorization plays a crucial role when dealing with polynomials as well, where we often seek factors of integer coefficients within the polynomial. However, the Factor Theorem aids in finding factors that are not immediately obvious.
In the original exercise, checking divisibility of \( n^3 - n + 3 \) by the integer 3 can be addressed with integer factorization methods. Yet, it is essential to recognize that in the context of polynomials, we are not simply looking for integer factors but rather polynomial factors. Seeing that the exercise asks to prove inequality, rather than just factorization, suggests students may need to consider how factorization could apply to inequalities.
Proof of Inequality
Inequalities are statements that suggest a relationship of less than, greater than, less than or equal to, or greater than or equal to between two expressions. Proving an inequality often involves showing that the relationship holds true for all values within a certain set. Unlike equations, inequalities do not suggest equality, but rather a range of possible values.
To prove an inequality, the Factor Theorem can sometimes be employed. For instance, if an exercise suggests that \( n^3 - n + 3 > 0 \) for all integer values of \( n \), then one way to approach the proof could be to find factors of the polynomial expression and analyze their signs over the set of integer values to see if the inequality consistently holds true.
Proofs of inequalities may require different steps than simple factorization. It might involve techniques such as the method of induction, direct calculation, or considering special cases. It's important for students to understand how the Factor Theorem can be implicated in these proofs and that proving an inequality demands a thorough verification for the defined range or set of numbers.
To prove an inequality, the Factor Theorem can sometimes be employed. For instance, if an exercise suggests that \( n^3 - n + 3 > 0 \) for all integer values of \( n \), then one way to approach the proof could be to find factors of the polynomial expression and analyze their signs over the set of integer values to see if the inequality consistently holds true.
Proofs of inequalities may require different steps than simple factorization. It might involve techniques such as the method of induction, direct calculation, or considering special cases. It's important for students to understand how the Factor Theorem can be implicated in these proofs and that proving an inequality demands a thorough verification for the defined range or set of numbers.
Other exercises in this chapter
Problem 38
In Exercises 35 - 38, you are given the probability that an event will happen. Find the probability that the event will not happen. \( P(E) = \dfrac{2}{3} \)
View solution Problem 38
In Exercises 19 - 40, use the Binomial Theorem to expand and simplify the expression. \( \left(\dfrac{2}{x} - 3y\right)^5 \)
View solution Problem 38
In Exercises 35 - 44, write an expression for the \( n \)th term of the geometric sequence. Then find the indicated term. \( a_1 = 64, r = -\dfrac{1}{4}, n = 10
View solution Problem 38
In Exercises 37-42, use a graphing utility to graph the first 10 terms of the sequence. (Assume that \( n \) begins with 1.) \( a_n = 2 - \dfrac{4}{n} \)
View solution