Problem 67 - Algebra and Trigonometry

Question

Suppose you were asked to solve the following two problems on a test: A. Find the remainder when \(6 x^{1000}-17 x^{562}+12 x+26\) is divided by \(x+1\) B. Is \(x-1\) a factor of \(x^{567}-3 x^{400}+x^{9}+2 ?\) Obviously, it's impossible to solve these problems by dividing, because the polynomials are of such large degree. Use one or more of the theorems in this section to solve these problems without actually dividing.

Step-by-Step Solution

Answer

Remainder of A: 3; B: No, \(x-1\) is not a factor.

Step 1: Understand the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by a linear divisor \( x - a \) is \( f(a) \). This means we can find the remainder by simply evaluating the polynomial at \( x = -1 \) for part A, and checking if \( f(1) = 0 \) for part B.

Step 2: Part A – Evaluate the Polynomial for Remainder

Given the polynomial \( 6x^{1000}-17x^{562}+12x+26 \), use the Remainder Theorem. Set \( x = -1 \) to find the remainder:\[6(-1)^{1000} - 17(-1)^{562} + 12(-1) + 26 \]Calculate each term:- \((-1)^{1000} = 1\) - \((-1)^{562} = 1\)So the polynomial becomes:\[ 6(1) - 17(1) + 12(-1) + 26 = 6 - 17 - 12 + 26 \]Simplify to find the remainder: \(3\).

Step 3: Part B – Check Factor Using the Factor Theorem

Given the polynomial \( x^{567} - 3x^{400} + x^9 + 2 \), use the Factor Theorem to determine if \( x - 1 \) is a factor. Set \( x = 1 \) and evaluate the polynomial:\[1^{567} - 3(1)^{400} + 1^9 + 2 = 1 - 3 + 1 + 2\]Calculate:- \(1^{567} = 1\)- \(1^{400} = 1\)- \(1^9 = 1\)Simplify the expression:\[ 1 - 3 + 1 + 2 = 1 \]Since the result is not zero, \( x-1 \) is not a factor.

Key Concepts

Factor Theorem

The Factor Theorem is a useful concept in algebra when working with polynomials. It states that a polynomial \( f(x) \) has a factor \( (x - a) \) if and only if \( f(a) = 0 \). This theorem helps us determine whether a given linear expression, like \( x-1 \), is a factor of a polynomial without performing the long division.For example, in the exercise, we used the Factor Theorem to check if \( x-1 \) is a factor of \( x^{567} - 3x^{400} + x^9 + 2 \). By evaluating the polynomial at \( x = 1 \), if the result is zero, it confirms \( x-1 \) is a factor. However, in our case, the evaluation gave us 1, meaning \( x-1 \) is not a factor.Understanding the Factor Theorem saves a lot of work, especially with high-degree polynomials. It can indicate possible roots of a polynomial, which are critical for factoring and solving polynomial equations. This theorem forms the basis for more complex algebra concepts and is essential for tackling problems in polynomial factorization.

Polynomial division

Polynomial division, though daunting at first, is a technique akin to long division for numbers. It allows us to divide one polynomial by another, typically a lower degree polynomial. However, with high-degree polynomials, the task can become extremely tedious, which is why theorems like the Remainder Theorem are such a blessing.When manually dividing polynomials, we aim to determine how many times the divisor, which is the polynomial we are dividing by, goes into the dividend, which is the polynomial we are dividing. The process involves multiple steps of division and subtraction, similar to numeric long division. While it's a systematic process, it quickly becomes impractical with high-degree polynomials due to the number of calculations involved.The exercise asked us to find a remainder after dividing a polynomial by \( x+1 \). Utilizing the Remainder Theorem simplified this potentially complex polynomial division by simply evaluating the polynomial at \( x = -1 \). This way, we avoided manual polynomial division, making the solution both quick and efficient.

Polynomial evaluation

Polynomial evaluation is a straightforward process that involves substituting a specific value for each variable in the polynomial and then calculating the result. This method is especially useful when applying the Remainder and Factor Theorems, as shown in the exercise.Evaluating a polynomial is critical in various scenarios:

• Checking the remainder of a polynomial after division by a linear factor, which can be swiftly done using the Remainder Theorem.
• Determining if a polynomial has a specified factor, utilizing the Factor Theorem.
• Calculating polynomial values at specific points, which is often needed in plotting graphs or simulations.

In the exercise, we evaluated the polynomial \( 6x^{1000} - 17x^{562} + 12x + 26 \) at \( x = -1 \) to find the remainder when divided by \( x+1 \). This evaluation method is efficient and allows us to solve problems involving high-degree polynomials with ease, demonstrating how evaluation serves as a cornerstone in handling polynomials effectively.