Problem 28 - Algebra 2

Question

Simplify. $$ \frac{3 c^{5}+5 c^{4}+c+5}{c+2} $$

Step-by-Step Solution

Answer

\(3c^4 - c^3 + 2c^2 - 4c + 9 - \frac{13}{c+2}\)

Step 1: Set Up Synthetic Division

To simplify the expression \( \frac{3c^5 + 5c^4 + c + 5}{c+2} \), we will use synthetic division. First, write down the coefficients of the numerator: \([3, 5, 0, 0, 1, 5]\). We need to include terms for all powers of \( c \) in descending order. Since there are no \( c^3 \) and \( c^2 \) terms, we insert 0s for those. Use \(-2\) as the divisor because we change the sign of the constant in the divisor \(c+2\).

Step 2: Perform Synthetic Division

Start with the leading coefficient 3. Drop it down below the line. Multiply 3 by -2, giving -6, and add this result to the next coefficient: \(5 + (-6) = -1\). Repeat this process with the new value: multiply, then add: 1. \(-1 \times -2 = 2\), add: \(0 + 2 = 2\)2. \(2 \times -2 = -4\), add: \(0 - 4 = -4\)3. \(-4 \times -2 = 8\), add: \(1 + 8 = 9\)4. \(9 \times -2 = -18\), add: \(5 - 18 = -13\)Thus, the result from synthetic division is: \([3, -1, 2, -4, 9, -13]\).

Step 3: Write the Quotient and Remainder

The coefficients \([3, -1, 2, -4, 9]\) represent the quotient, starting from the highest degree less than the original, \(3c^4 - c^3 + 2c^2 - 4c + 9\). The last number, \(-13\), is the remainder. Therefore, the expression becomes:\[3c^4 - c^3 + 2c^2 - 4c + 9 + \frac{-13}{c+2}\]

Step 4: Combine Terms and Finalize Simplification

The simplified form of the original expression, obtained by dividing, is:\[3c^4 - c^3 + 2c^2 - 4c + 9 - \frac{13}{c+2}\] This is the simplest form since the division has been completed, and the remainder is expressed over the original divisor.

Key Concepts

Polynomial Division

Polynomial division is a technique that involves dividing one polynomial by another. This is similar to the usual arithmetic division but with polynomials. When we have a polynomial expression, such as \(3c^5 + 5c^4 + c + 5\), and we need to divide it by \(c + 2\), we need a method that simplifies the division process. In this context, synthetic division becomes a helpful technique because it streamlines the process, especially when dealing with linear divisors of the form \(c + a\).
- **Setup:** Write down coefficients of the polynomial in descending order. If any term is missing from the sequence, use zero for its coefficient.- **Execute Synthetic Division:** Use the opposite of the constant term from the divisor. This becomes your synthetic divisor.- **Result:** You are left with the quotient and sometimes a remainder, simplifying the polynomial division process extensively.

Rational Expressions

A rational expression is a fraction where the numerator and the denominator are polynomials. In the given exercise, \( \frac{3c^5 + 5c^4 + c + 5}{c + 2} \) is a rational expression. Simplifying such expressions often requires polynomial division.
This transforms complex rational expressions into more understandable results with possibly simpler polynomials and a leftover remainder term if the divisor doesn’t divide exactly.
- **Normalization:** The primary aim when working with rational expressions is to express them in their simplest form. This simplification might include polynomial division, eliminating common factors if possible, or rewriting them to make analysis straightforward.- **Application:** Rational expressions are crucial in algebra, calculus, and other mathematical fields. They simplify expressions and make computation manageable, facilitating the solving of more complex algebraic equations and functions.

Remainder Theorem

The Remainder Theorem is a powerful concept in algebra that connects polynomial division with evaluation of the polynomial. According to the theorem, if a polynomial \( P(c) \) is divided by \( c - a \), the remainder of this division is \( P(a) \).
Using the Remainder Theorem in our exercise allows us to check if a given divisor is indeed a factor of the polynomial. If we find that \( P(-2) = 0 \), with \( c + 2 \) being our divisor, it affirms that there is no remainder, hence the divisor perfectly factors the polynomial.

• **Computation of Remainder:** The final value obtained in synthetic division represents the remainder, which can be checked using the theorem to ensure accuracy.
• **Verification:** Beyond simplifying polynomial expressions, the Remainder Theorem provides a quick verification mechanism for the accuracy of synthetic division or common factor identification.

Understanding the Remainder Theorem helps in predicting results of the division without performing the entire division process, saving time and effort in problem-solving.