Problem 8 - College Algebra

Question

For the following exercises, use the Remainder Theorem to find the remainder. $$ \left(x^{4}+5 x^{3}-4 x-17\right) \div(x+1) $$

Step-by-Step Solution

Answer

The remainder is \(-17\).

Step 1: Identify the Polynomial and Divisor

We are given the polynomial \( f(x) = x^4 + 5x^3 - 4x - 17 \) and asked to find the remainder when it is divided by \( x + 1 \).

Step 2: Apply the Remainder Theorem

The Remainder Theorem states that the remainder of the division of a polynomial \( f(x) \) by \( x-c \) is \( f(c) \). For the divisor \( x+1 \), we have \( c = -1 \).

Step 3: Evaluate the Polynomial at c

Substitute \( -1 \) into the polynomial: \( f(-1) = (-1)^4 + 5(-1)^3 - 4(-1) - 17 \).

Step 4: Calculate the Value

Calculate each term: \((-1)^4 = 1\), \(5(-1)^3 = -5\), \(-4(-1) = 4\). Add these results: \( 1 - 5 + 4 - 17 = -17 \).

Step 5: Conclusion

The remainder when \( x^4 + 5x^3 - 4x - 17 \) is divided by \( x + 1 \) is \(-17\).

Key Concepts

Polynomial Division

Polynomial division is a method used to divide one polynomial by another. It is similar to long division of numbers. When performing polynomial division, you find out how many times the divisor fits into the polynomial, and determine a quotient and a remainder. The polynomial you start with is called the dividend, while the polynomial you are dividing by is the divisor. In our original exercise, the polynomial

• Dividend is \( x^4 + 5x^3 - 4x - 17 \)
• Divisor is \( x + 1 \)

Understanding polynomial division is key to working with the Remainder Theorem, as it helps us determine how any leftover portion, or remainder, fits into our calculations when dividing polynomials.

Evaluate Polynomial

To evaluate a polynomial means to substitute a specific value in place of the variable in the expression and then simplify to find the result. This is a critical step in using the Remainder Theorem.
In the exercise, we substitute \(-1\) into the polynomial \( f(x) = x^4 + 5x^3 - 4x - 17 \) to find \( f(-1) \).
Evaluating a polynomial involves:

• Replacing each \( x \) in the polynomial with the given number (\(-1\) in this case).
• Simplifying the expression by calculating the powers of the number and then carrying out any additions and subtractions.

This process helps determine the value or the remainder when dividing by a binomial such as \( x+1 \).

Algebraic Expression

An algebraic expression involves numbers, variables, operations (like addition, subtraction, multiplication), and exponents. It's essential in forming both polynomials and equations.
In our exercise, the polynomial \( x^4 + 5x^3 - 4x - 17 \) is an algebraic expression.
Characteristics include:

• Each term in a polynomial is separated by the addition or subtraction sign.
• Variables can appear with exponents, like \( x^4 \) or \( x^3 \).
• The degree of the polynomial is defined by the highest power of its variable, which is 4 in our case.

Recognizing these characteristics helps in managing the expressions correctly during operations like division or evaluation.

Synthetic Division

Synthetic division is a simplified form of polynomial division, particularly when the divisor is a binomial in the form of \( x-c \). This method is much simpler and quicker than long division, as it reduces the amount of space and calculations required.
While synthetic division is not used directly in evaluating the problem without changing the original setup, understanding it can help confirm the remainder obtained using the Remainder Theorem.

• It involves using only the coefficients of the polynomial.
• You perform a series of multiplication and addition operations.
• The final row in the calculation gives you the result and remainder.

This technique complements what we do with the Remainder Theorem and serves as a verification method for re-checking our solutions or understanding division results.